Prison Shape and Sunlight
Chapter Title : Prison Shape and Sunlight
Before reading this chapter you should already be familiar with the following
Algebra
Trigonometry
Cartesian Coordinates
Polar Coordinates
How to convert degrees to radians
The connection between radians and the lengths of curved objects that are shaped like parts of a circle
How to use sine and cosine to measure the length of objects given an angle and another length
How to use inverse tangent to figure out an angle given two perpendicular lengths or a slope of a line
Understand vectors at a high school physics level or college physics level without calculus or better
You should understand the concept of what a vector perpendicular to a plane would be
You should understand how to take limits in calculus
What you do not need to understand this material is any high school or college geometry class material that is not covered in algebra, trigonometry, calculus or physics
Notation
* means multiply or times or multiplication sign
/ means divide by or division sign
x ^ N means x to the N power
x ^ 2 means x squared
x ^ 0.5 means the square root of x
Pi means pie or approximately 3.14
The full words sine, cosine and tangent are written instead of sin, cos and tan to make it easier to hear for transcribing into audiobook format and devices for the blind to hear electronic text
cosine ^ 2 ( x) means cosine ( x ) * cosine ( x )
sine ^ 2 ( x) means sine ( x ) * sine ( x )
tangent ^ 2 ( x) means tangent ( x ) * tangent ( x )
>= means greater than or equal to
<= means less than or equal to
https://web.archive.org/web/20221107183232/https://en.wikipedia.org/wiki/Relational_operator
Perimeter arc length or path length of a section of a circle = Radius * Angle in Radians
Angle on Radians = Perimeter arc length or path length of a section of a circle / Radius
Modeling light
I will explain physics concepts how the angle of incidents and inverse square law effect light brightness. I will later use this to discuss seasons and sunrise and sunset in flat earth models vs globe models. The equations are very simple but my explanation of where the equations might come from will be more complicated. There also is a question which shall be addressed of do the angle of incidence equations regarding intensity violate the conservation of energy and should they be used at all?
Intensity or flux?
Intensity is a type of flux
I have chosen to use the term intensity instead of flux because it is more specific
I am concerned if I used the term flux instead of intensity in all cases this could potentially be misleading because someone might confuse it with one of the other many meanings of flux
It might have been better for me to use the term flux instead of intensity in some cases if the meaning of the term intensity is too specific for the way in which I used it to qualify as a correct use of the term intensity
What is isotropy?
"An isotropic radiator is a theoretical point source of electromagnetic or sound waves which radiates the same intensity of radiation in all directions. It has no preferred direction of radiation. It radiates uniformly in all directions over a sphere centred on the source."
https://web.archive.org/web/20221019153807/https://en.wikipedia.org/wiki/Isotropic_radiator
What is rotational symmetry?
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn
https://web.archive.org/web/20221211180414/https://en.wikipedia.org/wiki/Rotational_symmetry
Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the nth order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of 360°/n (180°, 120°, 90°, 72°, 60°, 51 3⁄7°, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all objects look alike after a rotation of 360°).
https://web.archive.org/web/20221211180414/https://en.wikipedia.org/wiki/Rotational_symmetry
51 3⁄7 means 360 / 7 in wikipedia quote above
Three dimensional isotropic light models
Rules for ray orientation
Requirement for arrangement of rays originating from a source. Each ray represents the travel path of a photon emitted from a point source. This is the travel path a photon would experience if it is not reflected, absorbed or refracted and does not collide and interact with any other particles while traveling. This is for representing a light source that is isotropic or almost isotropic in three dimensions.
Imagine a finite or unlimited number of rays originating from a specific point in different directions
If the number is finite then the source is not truly isotropic but can be treated as close enough to isotropic if these requirements are met
If a plane that intersects with that point is oriented in any direction there should be an equal number of rays on each side the plane divides in half.
If a plane that intersects with that point is oriented in any direction there should be at least one ray on each side the plane divides in half.
These halves do not need to be mirror images
When counting the number of rays on each side the plane divides if a ray is parallel to the plane then it will be counted toward the total on both sides and not toward the total on only one side. Or alternatively it will not be counted toward the total at all. Either way such rays parallel to the plane will not effect the outcome of whether or not each half has an equal number of rays.
Axes is the plural of axis
There will exist at least three different axes through which the orientation of the rays will be radially symmetric. All these axes will go through the source point.
There will be a axis going through that point which can be oriented such that the orientation of the rays has radial or rotational symmetry
There will be two other axes perpendicular to that axis and also perpendicular to each other which can be oriented in such a way that the rays have radial or rotational symmetry along these axes also.
One way to meet these requirements for a finite and limited number of rays is for each ray to be oriented perpendicular to the face of a type of regular polyhedron called a platonic solid and for each ray to originate at the center of this platonic solid
However it is best to assume there are an unlimited number of such rays.
The arrangement of the ray's would like a spiky ball. Someone might have an argument about if an unlimited number of such spikes would form a sphere or if there could be an unlimited number of spikes with an unlimited number of gaps between spikes. Such an argument might miss the main point I am trying to represent but I have foreseen it.
"Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids"
https://web.archive.org/web/20221211180414/https://en.wikipedia.org/wiki/Rotational_symmetry
https://web.archive.org/web/20221225023121/https://www.pinterest.com/pin/spiky-ball-detail--391039180142444740/
Requirements for the shape of a surface surrounding such rays
The surrounding surface is not a physical object but a imaginary mental construction used to calculate how many rays or photons would pass through per surface area at a region in space if none of the photons disappeared due to interaction with a real physical object or medium before reaching the region.
Such a surface must be shaped such that each ray will only intersect with the surface exactly one time
There must be no rays that do not intersect the surface
There must be no rays that intersect with the surface two or more times
The surface must not be parallel with a ray it intersects at a point of intersection
Such a surface must surround the light source
The surface must be shaped such that as long as the rays meet the rules for orientation all those rules hold no matter how the rays are oriented or how many rays there are
Such surfaces could include but are not limited to parallelepipeds and spheroids
Some more common shape names that would work include a box shape, a rectangular solid, a sphere or a cylinders
An example of a shape that would not work to meet the requirements would be a donut shape with a hole in the middle
Light and the inverse square law in three dimensions
For a number of photons that originate from a isotropic light source
As long as the shape requirements and light ray orientation requirements and lack of collision and absorption requirements are met then the following applies
The number of photons per surface area * The surface area = The original number of photons from the source
No matter how you change the shape and size of the surface as long as the shape requirements are met the same number of photons will pass through the entire surface area. But the number of photons per surface area can change
If you multiply the radius of a sphere by N then it's surface area will be multiplied by N squared
The photons per surface area that go through a spherical surface surrounding a light source will be inversely proportional to the radius of the sphere squared
If you multiply the side length of a cube by N then the surface area of the cube will be multiplied by N squared
The photons per surface area that go through a cube surface will also be inversely proportional to the side length of the cube squared
If the light source is in the exact center of the sphere then for any two different sections of the same sphere with equal surface area the same number of photons will go through
If the light source is in the exact center of the cube then for any two different sections of the same cube with equal surface area there is no guarantee that the same number of photons will go through unlike for the sphere. But none the less the same number of photons will go through the entire cube regardless of it's size.
If the source emits light then the quantity the source emits could be energy per time, photons per time, lumens of brightness or something else. If the quantity emitted is lumens of brightness than the brightness per area will be measured in lux. This principle can be used to calculate quantities of other things than light emitted by sources.
Quantity Emitted by source = Surface Area Surrounding Source * Quantity per surface area
This equation is versatile and can be used to understand things other than intensity at a single point but right now I will explain how to do so for the intensity of light at a single point with the inverse square law
Because of the uneven distribution of intensity across other shapes a imaginary sphere will be used to calculate intensity at a point
Quantity per surface area at a single point is as follows when the rays are perpendicular or "normal" to the surface they hit according to the inverse square law
Quantity emitted by source / Surface Area of a sphere with a radius equal to the distance from the source
Quantity emitted by source / ( 4 * pi * distance from source ^ 2 )
This is for isotropic sources when the emitted quantity continues to travel in straight directions ( plural ) from the source without interference forever at a constant speed without stopping. Failure to meet those requirements might mean the equation does not apply. This specific inverse square law equation is only meant to be applied to a single point and not to a large region. The equation is only meant for when the source emits photons ( or some quantity of something else ) in all three dimensions and not when the light ( or other emitted quantity ) is limited to 1 or 2 dimensions.
2 Dimensional Isotropic Light Models
Rules for ray orientation
Requirement for arrangement of rays originating from a source. Each ray represents the travel path of a photon emitted from a point source. This is the travel path a photon would experience if it is not reflected, absorbed or refracted and does not collide and interact with any other particles while traveling. This is for representing a light source that is isotropic or almost isotropic in two dimensions for which the ray does not travel in a third dimension.
Let there be three perpendicular or orthogonal dimensions in a Cartesian Coordinate System
The component of the vector of the light ray velocity in exactly one of the three dimensions is zero for all right rays
For the other two dimensions the rays have radial or rotational symmetry around a single axis
There can be a unlimited or finite number or rays
There are at least three rays
Requirements for the 2 dimensional outline of the shape surrounding such rays
The shape must surround or enclose the light source. The light source must be inside the shape.
Each ray must intersect with the shape exactly once. The shape must be such that each ray intersects with the shape exactly one time. This must be true no matter how the rays are positioned or how many rays there are as long as the rays follow the rules for there orientation.
Each ray must not intersect with the ray more than once. The shape must be such that no ray intersects with the shape more than once
There must be no rays that do not intersect with the shape. The shape must be such that no ray fails to intersect with it.
No ray must be parallel to any part of the shape at the specific location where the ray intersects with the shape
Thickness in the third dimension at which the light rays have a zero velocity component
Thickness Model 1
The third dimension is completely ignored and the thickness is ignored or considered negligible
Thickness Model 2
Both the rays and the shape have an equal thickness in that specific third dimension
Thickness Model 3
There would be planes perpendicular to a single axis. If there are N planes, the distance between each plane is
Thickness / ( N - 1 )
For example if there was a 1 inch thickness vertically and there were three planes then the planes would be 0.5 inches apart. If the bottom plane is at 0 inches height then the other two planes would be at 0.5 inches height and 1 inch height.
In each of these planes there would be an equal number of rays
Each plane would have at least three rays. There would be at least 3 * N rays total between all planes combined.
In this case there would still be rotational symmetry around exactly one axis
The rays in each plane would have rotational symmetry around the same one axis
Each of those planes would be perpendicular to that one and only axis
Intensity when isotropic in 2 dimensions
As long as the shape requirements and light ray orientation requirements and lack of collision and absorption requirements are met then the following applies
No matter how you change the shape and size of the surface as long as the shape requirements are met the same number of photons will pass through the entire surface area. But the number of photons per surface area can change
The number of photons per surface area * The surface area = The original number of photons from the source
The number of photons per length * The perimeter length = The original number of photons from the source
If the light source is in a circle then if the radius of the circle is multiplied by N then the perimeter surrounding the source will be multiplied by N
Circle Perimeter = 2 * pi * radius
If the light source is in a square then if the side length of the square is multiplied by N then the perimeter surrounding the source will be multiplied by N
Square Perimeter = 4 * side length
If the light source is in the exact center of the circle then for any two different sections of the same circle with equal perimeter or arc length the same number of photons will go through
If the light source is in the exact center of the square then for any two different sections of the same square with equal perimeter there is no guarantee that the same number of photons will go through unlike for the circle. But none the less the same number of photons will go through the entire square regardless of it's size.
Because the intensity is distributed unequally across other shapes a circle will be used to calculate intensity at a single point. This only applies when the surface the rays hit is perpendicular or "normal" to the rays.
Intensity = Quantity Emitted / ( 2 * pi * radius * thickness )
Model 1 ignoring thickness
Quantity per length = Quantity emitted / Perimeter of shape
Model 2 and Model 3
Quantity per surface area = Quantity emitted / Surface Area
Surface Area = Perimeter of Shape * Thickness of shape
Quantity per surface area = Original Quantity emitted / ( Perimeter of Shape * Thickness of Shape )
2 dimensional light rays over a limited angle beam with isotropy over that range and no rays outside the limited angle
Orientation of rays
Let us call the angle the rays are limited to by the name Theta
0 degrees < Theta < 360 degrees
There can be a finite or unlimited number of rays
In order to form a angle there need to be at least 2 rays
If there are exactly 2 rays then Theta can not equal 180 degrees. If there are 3 or more rays then Theta can equal 180 degrees or some other angle value
There are N rays
The nearest ray or pair of rays next to each ray is the following angle apart
Theta / ( N - 1 )
If for example Theta = 45 degrees and there are 4 rays then they might be oriented as follows
45 degrees / ( 4 -1 ) = 15 degrees
Ray 1 at ( 0 + x ) degrees
Ray 2 at ( 15 + x ) degrees
Ray 3 at ( 30 + x ) degrees
Ray 4 at ( 45 + x ) degrees
Ray 4 angle - Ray 1 angle = ( 45 + x ) degrees - ( 0 + x ) degrees = 45 degrees
Curve segment or line segment shape requirements
The curve segment or line segment must have two end points
The reason for having two end points is so that you can divide by the path length of the curve to calculate intensity without dividing by too large a quantity
Each end point of the curve segment or line segment must be intersected with by exactly one ray at it's exact location
Each ray must intersect with a curve segment or line segment exactly one time
A ray must not intersect with a curve segment or line segment more than one time
There must not be any rays that do not intersect with a curve segment or line segment
At the point of intersection between a curve segment or line segment and a ray the ray must not be parallel to that part of the curve segment or line segment
Thickness in the third dimension at which the light rays have a zero velocity component for the 2 dimensional case with a limited angle beam
Thickness Model 1
The third dimension is completely ignored and the thickness is ignored or considered negligible
Thickness Model 2
Both the rays and the shape have an equal thickness in that specific third dimension
Thickness Model 3
There would be planes perpendicular to a single axis. If there are N planes, the distance between each plane is
Thickness / ( N - 1 )
For example if there was a 1 inch thickness vertically and there were three planes then the planes would be 0.5 inches apart. If the bottom plane is at 0 inches height then the other two planes would be at 0.5 inches height and 1 inch height.
In each of these planes there would be an equal number of rays
Each plane would have at least two rays. There would be at least 2 * N rays total between all planes combined.
In this case there would still be rotational symmetry around exactly one axis
The rays in each plane would have rotational symmetry around the same one axis
Each of those planes would be perpendicular to that one and only axis
Intensity
The number of photons per length * The path length = The original number of photons from the source
The number of photons per surface area * The surface area = The original number of photons from the source
The intensity at a specific point where a ray is perpendicular or "normal" to the section of the line segment or curve it intersects with is
If Theta is in degrees then
Intensity = Quantity Emitted / ( Distance from source * Thickness * Theta * 2 pi radians / 360 degrees )
If Theta is in radians then
Intensity = Quantity Emitted / ( Distance from source * Thickness * Theta )
Model 1 ignoring thickness
Quantity per length = Quantity emitted / Path Length
Model 2 and Model 3
Quantity per surface area = Quantity emitted / Surface Area
Surface Area = Path Length * Thickness
Quantity per surface area = Original Quantity emitted / ( Path Length * Thickness )
Angle of incidence
Do not confuse the angle of incidence with other angles
The angle of incidence is the angle of the light ray relative to a plane perpendicular or normal to the section of the surface it hit's
The angle of incidence is not the angle of the light ray relative to a plane tangent or parallel to the section of the surface it hit's
If a ray is perpendicular to the section of the surface it hit's then it's angle of incidence is 0 degrees
Angle relative to surface refers to the shortest angle relative to a plane tangent to a surface. There is a second such angle at 180 degrees minus the first such angle.
Angle of incidence + Angle Relative to Surface = 90 degrees
Explanation of the relationship between angle of incidence and intensity of the brightness of light
Accessed online 2022 December 22
Intensity at the surface changes with the Cosine of the angle made with the Normal (perpendicular)
https://www.quora.com/How-are-the-light-angle-and-light-intensity-correlated
Angle of incidence and intensity of light on different sections of a cube
If a isometric light source is inside the center of a cube then at a corner of a cube the light will be at a lower intensity than at the center of a side of a cube, which will have a lower intensity than at a center of a face of a cube based on the inverse square law alone before considering any other factors.
If this cube was actually a closed physical cardboard box instead of a imaginary structure and a light source was emitting light at a constant amount of power or energy per time one would expect the light to heat up the center of a face of the box more than a center of a side of the box and to heat up a center of a side of the box more than the corner of the box. This is because each of those locations are at different distances and not because of the angle the light hits those locations.
At the center of the face of the box the angle of incidence is 0 degrees
As the light shines on different locations on a face of the box getting closer and closer to the center of the side of the box the angle of the incidence increases until it approaches 45 degrees as the location approaches the center of a side of the box
As the angle of incidence decreases the light heats up the box more per time
As the angle of incidence increases the light heats up the box less per time
Some people claim changing the angle of incidence changes the intensity of light but using a cube box for such a case would be a poor example. Because one would ask did the intensity really change because of an additional factor when the angle of incidence changed or only because the distance changed. Could it be possible that the angle of incidence did not matter at all. Is this a mistake where the distance causes the change in intensity and the angle of incidence does not cause the change in intensity but the distance is correlated with the angle of incidence?
Someone might think that if the box is closed then the power per area at each spot ( based on the inverse square law ) times the surface area of that spot for all the spots added together in the entire box should total up to the power before the factor of the angle of incident is ever considered.
If the intensity in some spots is reduced based on the angle of incidence in addition to the inverse square law alone then that person might think that the total energy from the photons hitting all the spots on the box added together will be less than the total energy emitted from the light source which might seem to them to violate the conservation of energy.
Someone could make arguments about the direction of the "momentum" of the light hitting a section of the cardboard box changing as the angle of incidence changes. But conservation of energy is a very different issue than conservation of momentum. A single photon at a single frequency should add the same amount of energy to it's target as the energy it took to produce the photon no matter what angle it hits it's target or else it violates the conservation of energy. Presuming the photon disappears and is not replaced or reflected after the collision. Sometimes the direction and magnitude of the "momentum" of light is important in particle physics such as the Compton and reverse Compton effect. I do not think knowing the formula for the "momentum" of light would help address the issue of angle of incident and conservation of energy.
If there really is a difference between light intensities at the same distance from the source when the angle of incidence changes then I would guess that the answer has something to do with surface areas.
Note : This is not the same way momentum is normally calculated for macroscopic or large size objects people normally interact with in their daily lives
p = magnitude of "momentum" of light
f = frequency
c = speed of light
p = E/c = hf/c = hf/λf = h/λ
p = h / λ
Where:
P: photon momentum
h:Plank's constant
λ: photon's wavelength
https://web.archive.org/web/20171024143614/https://www.softschools.com/formulas/physics/photon_momentum_formula/542/
https://web.archive.org/web/20221125080227/https://en.wikipedia.org/wiki/Compton_scattering
https://web.archive.org/web/20221006013710/https://en.wikipedia.org/wiki/File:Compton-scattering.svg
https://web.archive.org/web/20220701130837/https://unitednuclear.com/index.php?main_page=product_info&cPath=29_55&products_id=1312
https://web.archive.org/web/20221014022335/https://unitednuclear.com/bmz_cache/t/t-shirts-shirt_black_mockup_compton_wbpng.image.550x550.png
Straight Outta Compton is a 2015 American biographical drama film directed by F. Gary Gray, depicting the rise and fall of the hip hop group N.W.A and its members Eazy-E, Ice Cube, Dr. Dre, MC Ren, and DJ Yella.
https://web.archive.org/web/20221015022151/https://en.wikipedia.org/wiki/Straight_Outta_Compton_(film)
http://web.archive.org/web/20221228231939/https://en.wikipedia.org/wiki/Straight_Outta_Compton_(disambiguation)
The key question
Something I really want to know about angles of incidence is if the distance between the light source and the target remains the same then will the change in the angle of incidence decrease the intensity at the target location or will the intensity stay the same?
Scenario 1
The target of a light source is a board of unspecified width and thickness with a specified 2 meter length located between coordinates ( -1 meters, 0 ) and ( 1 meters, 0 )
the center of the target is therefore located at ( 0, 0 )
The light source is located at ( x meters, 1 meter )
the angle of incidence = 90 degrees - inverse tangent ( 1 / x )
x = tangent ( 90 degrees - angle of incidence )
based on the inverse square law alone the intensity at the location of the center of the board is
constant / ( 1 meter ^ 2 + x ^2 meters ^ 2 )
constant / ( 1 meter ^ 2 + tangent ( 90 degrees - angle of incidence ) ^2 )
This is ignoring any additional modification from the angle of incidence
In this case as x increases the distance of the center of the object from the source increases and the intensity at that location decreases based on the inverse square law. But as x increases the angle of incidence also increases which also is claimed to decrease intensity. In this case the intensity is correlated with the distance away from the source and the distance away from the source is correlated with the angle of incidence. If an increased angle of incidence and a decreased intensity are both caused by a increased distance away from the source then increasing the angle of incidence might not decrease the intensity of light when the distance from the source is held constant.
It is therefore difficult in this scenario to evaluate if the change in angle of incidence changed the intensity or if that was really only caused by the change in distance from the source alone.
If the formula for parallel rays at a long distance was used then if the angle of incidence was considered in addition to the inverse square law then the intensity would be the following with to the two formulas combined. Although this might or might not be too close a distance for the rays to be treated as effectively parallel.
constant * cos ( angle of incidence ) / ( 1 meter ^ 2 + x ^2 meters ^ 2 )
constant * cos ( angle of incidence ) / ( 1 meter ^ 2 + tangent ( 90 degrees - angle of incidence ) ^2 )
Scenario 2
The target of a light source is a board of unspecified width and thickness with a specified 2 meter length located between coordinates ( -1 meters, 0 ) and ( 1 meters, 0 )
the center of the target is therefore located at ( 0, 0 )
The light source is located at ( x meters, y meters )
where x^2 + y^2 = 1
If the light source is at a constant distance of 1 meter away from the board then the intensity at the location of the center of the board based on the inverse square law alone would be constant and the same for any angle selected. This is because the distance the light source is away from the center of the board does not change when the angle of incidence changes in this scenario unlike the previous scenario.
angle relative to surface = inverse tangent ( y / x )
the angle of incidence = 90 degrees - inverse tangent ( y / x )
The intensity adjusted for the angle of incidence and the inverse square law combined would be the following based on the formula for parallel rays at a long distance. Although this might or might not be too close a distance to treat the rays as effectively parallel.
constant * cosine ( angle of incidence )
constant * cosine ( 90 degrees - inverse tangent ( y / x )
constant * sine ( angle relative to surface )
constant * sine ( inverse tangent ( y / x ) )
Relation between intensity, surface area and number of rays
The intensity equals a constant times the number of rays per surface area
Intensity = constant / length per ray
Intensity = constant * number of rays / length
Surface Area = Path Length * Thickness
Intensity * Surface Area = Quantity Emitted
Intensity at 0 degrees angle of incidence * Surface area at 0 degrees angle of incidence = Intensity at Beta degrees angle of incidence * Surface area at Beta degrees angle of incidence
Intensity at Beta degrees angle of incidence = Intensity at 0 degrees angle of incidence * Surface area at 0 degrees angle of incidence / Surface area at Beta degrees angle of incidence
Surface Area at 0 degrees angle of incidence / Surface Area at beta degrees angle of incidence = Path Length at 0 degrees Angle of incident * thickness / ( Path length at Beta degrees angle of incidence * thickness ) = Path length at 0 degrees angle of incident / path length at Beta degrees angle of incidence
Two different models for counting the number of rays per length or length per ray
Two different models can be done to count the length per ray or number of rays per length each giving a different result for a finite number of rays. But both models give the same result for an unlimited number of rays. At least from the viewpoint in which 0.9 is the same as 1. Which is effectively the same for all practical purposes after rounding to as close an amount as you want no matter how small that amount is.
https://web.archive.org/web/20221222121202/https://en.wikipedia.org/wiki/0.999...
Model 1 using the length between two closest rays
If all rays hit a target at y coordinate equals zero
But ray 1 hit the light source at x = 0 meters
ray 2 hit the light source at x = 1 meter
ray 3 hit the light source at x = 2 meters
If these same coordinates given were used for any three rays in both model 1 and model 2 then a ray would hit the light source once per meter. This is two thirds times the number of rays per meter as in model 2. This is one and a half times the number of meters per ray as in model 2
Length per ray = Total length / ( Number of rays -1 )
Rays per length = ( Number of rays - 1 ) / Length
Model 2 using the length divided by the number of rays
If all rays hit a target at y coordinate equals zero
But ray 1 hit the light source at x = 0 meters
ray 2 hit the light source at x = 1 meter
ray 3 hit the light source at x = 2 meters
Rays per length = Number of rays / Total Length
Length per ray = Total Length / Number of rays
If these same coordinates given were used for any three rays in both model 1 and model 2 then a ray would hit the light source three times every two meters. This is two thirds times the number of meters per ray as in model 1. This is one and a half times the number of rays per meter as in model 1.
Reconciling model 1 and model 2 for a unlimited number of rays
Limit as n approaches positive infinity rays of
total length / ( n - 1 ) = total length / ( n )
Angles of incidence and a light source at a long enough distance to treat rays as parallel
Let the flat surface of the object hit by light lie purely horizontally so that a vector with an angle of incidence of zero degrees is purely vertical
Inensity at Beta radians angle of incidence / intensity at 0 radians angle of incidence = horizontal distance between rays at 0 radians angle of incidence / horizontal distance between rays at beta radians angle of incidence
SURF = Angle of ray relative to surface
AINC = Angle of incidence
SURF + AINC = 90 degrees
Sine ( SURF ) = Cosine ( AINC )
Cosine ( SURf ) = Sine ( AINC )
Hor = The shortest Horizontal Distance between two rays
Short = The Shortest Distance between two rays
Vert = the shortest Vertical Distance between two rays
If AINC = 0 degrees then Short = Hor and Vert = 0
When 0 degrees >= AINC < 90 degrees
Y1 = Y coordinate of ray 1
Y2 = Y coordinate of ray 2
X1 = X coordinate of ray 1
X2 = X coordinate of ray 2
Direction of Short Vector is perpendicular to both Ray 1 and Ray 2
Magnitude of Short Vector = Short
Short Vector = ( Short * cosine ( SURF + 90 degrees ) , Short * sine ( SURF + 90 degrees ) )
X1 = K * cosine ( SURF )
Y1 = K * sine ( SURF )
X2 = X1 + Short * cosine ( SURF + 90 degrees )
Y2 = Y1 + Short * sine ( SURF + 90 degrees )
90 degrees = 0.5 * pi radians
cosine ( Theta + 90 degrees ) = - sine ( Theta )
sine ( Theta + 90 degrees ) = cosine ( Theta )
https://web.archive.org/web/20221211225057/https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Shifts_and_periodicity
X2 = K * cosine ( SURF ) - short * sine ( SURF )
Y2 = K * sine ( SURF ) + short * cosine ( SURF )
Solve for X2 when Y2 equals zero to find Hor
0 = K * sine ( SURF ) + short * cosine ( SURF )
K = - short * cotangent ( SURF )
X2 = [ - short * cotangent ( SURF ) * cosine ( SURF ) ] - short * sine ( SURF )
X2 = - short * [ cotangent ( SURF) * cosine ( SURF ) + sine ( SURF ) ]
X2= - short *[ cotangent ( AINC - 90 degrees ) * cosine ( AINC - 90 degrees ) + sine ( AINC - 90 degrees) ]
X2 = - short * [ tangent ( AINC ) * sine ( AINC ) + cosine ( AINC ) ]
Hor = - X2 when Y2 equals zero
Hor = short * [ tangent ( AINC ) * sine ( AINC ) + cosine ( AINC ) ]
Hor = Short * [ Cosine ( AINC ) ^ 2 + Sine ^ 2 ( AINC ) ] / Cosine ( AINC)
Sine ^ 2 ( theta ) + cosine ^ 2 ( theta ) = 1
Hor = Short / Cosine ( AINC )
H1 + H2 = Hor
V1 + V2 = Vert
V1 = Short * sine ( SURF + 90 degrees ) = Short * cosine ( SURF )
H2 = Short * cosine ( SURF + 90 degrees ) = Short * sine ( SURF )
H1 = V1 / Tangent ( SURF ) = V1 * Cotangent ( SURF )
H1 = Short * cosine ( SURF ) * Cotangent ( SURF )
H1 + H2 = Short * [ cosine ( SURF ) * Cotangent ( SURF ) + Sine ( SURF ) ]
Hor = H1 + H2 = Short * [ Sine ( AINC ) * Tangent ( AINC ) + Cosine ( AINC ) ]
Hor = Short * [ Cosine ( AINC ) ^ 2 + Sine ^ 2 ( AINC ) ] / Cosine ( AINC)
Sine ^ 2 ( theta ) + cosine ^ 2 ( theta ) = 1
Hor = Short / Cosine ( AINC )
Intensity ( AINC = 0 radians ) / Intensity ( AINC ) = Hor ( AINC ) / Hor ( AINC = 0 radians )
Intensity ( AINC = 0 radians ) / Intensity ( AINC ) = Short / [ Short / Cosine ( AINC ) ]
Intensity ( AINC = 0 radians ) / Intensity ( AINC ) = cosine ( AINC )
Intensity ( SURF = 0.5 * pi radians ) / Intensity ( SURF ) = sine ( SURF )
Diagram shows Ray 2 located above or to the left of Ray 1.
Template made in geogebra edited with image flip to produce other picture
Angles of incidence and intensity of a light source at to short a distance to treat rays as parallel
These are light source problems that are more complicated because the rays are not modeled as parallel. These light source problems sometimes do not result in exactly the same intensity formula for angle of incidence when the rays are close to parallel. Sometimes the intensity formula is very close to the formula for when the rays are treated as parallel.
Explaining angle of incidence based on a beam of light of limited angle whose middle most ray is a constant distance from the center of the target having a changing surface area as the angle of incidence changes. Using two different coordinate system models.
Coordinate System model 1
Using Geogebra. Based on a coordinate system in which the coordinates of the light source change and the coordinates of the object hit by the light source stay the same.
In this model I created a line segment that goes through the coordinate ( 0 , 0 ) and rotated it 15 degrees to find information in the method I shall describe.
The "roots" on Geogebra were presumed to mean where the x coordinate of a function intersects the x axis where the y coordinate is equal to zero. When the "roots" listed a y coordinate of zero in Geogebra.
Since Geogebra did not have me put in units for the coordinates I had it have a unitless length of 2. The unitless length was 2 instead of 1 because if the length was 1 when it was rotated it would not intersect with the x axis. The line segment went from the coordinates of the light source divided by 1 meter to the opposite coordinates of the light source divided by 1 meter.
The length was calculate using the x coordinates found in the "roots" from Geogebra of a 15 degree rotation in each direction of clockwise and counterclockwise for opposite direction rotations of the same line segment
This length was then compared with another formula to calculate the length explained in coordinate system model 2 and found to be the same up to 6 decimal places
The relative intensity at a specific angle compared to a SURF angle of 90 degrees or AINC angle of 0 degrees was calculated by dividing the length at SURF angle of 90 degrees by the length at that specific angle
Abbreviations
SURF = Angle relative to surface of the middle ray
AINC = Angle of incidence of the middle ray
SURF = 90 degrees - Angle of incident
Problem's Parameters
The length of the object is 200 meters but the length. width and thickness of the object does not matter for the calculations in this model so long as it is larger than the length the light source hits. A length was arbitrarily assigned so I could give coordinates for the center of the object.
Center of object in cartesian coordinates = ( 0 meters, 0 meters )
Object coordinates ( - 100 meters, 0 meters) to ( 100 meters, 0 meters )
Distance center of object is away from source = 1 meter
Number of rays = 3
The middle ray hits the center of the object
The middle ray is ray 2
Limited Angle of light source beam = 30 degrees
Light Source Coordinates = ( - cosine ( SURF ) , sine ( SURF) ) meters
Light Source Coordinates = ( - sine ( AINC ) , cosine ( AINC ) ) meters
Length is the length from where Ray 1 hits the object to where where Ray 3 hits the object
All rays hit object at y Coordinate = 0
Ray x Coordinates means where that Ray hits the object
Ray 2 x Coordinates = 0 for all SURF angles
Examples calculated in Geogebra with pictures
Results were rounded truncated to 6 decimal places
For SURF 90 degrees and AINC 0 degrees
Ray 1 x Coordinate in meters = - 0.267949
Ray 3 x Coordinate in meters = + 0.267949
Length in meters = 0.535898
cosine ( AINC) * [ Tangent ( AINC + 15 degrees ) - Tangent ( AINC - 15 degrees ) ] = 0.535898
Length at Surf 90 degrees / Length = 1
Sine ( Surf ) = Cosine ( AINC ) = 1
For SURF 75 degrees and AINC 15 degrees
Ray 1 x Coordinate in meters = -0.258819
Ray 3 x Coordinate in meters = 0.298858
Length in meters = 0.557677
Length at Surf 90 degrees / Length = 0.960946
cosine ( AINC) * [ Tangent ( AINC + 15 degrees ) - Tangent ( AINC - 15 degrees ) ] = 0.557677
Sine ( Surf ) = Cosine ( AINC ) = 0.965925
For SURF 60 degrees and AINC 30 degrees
Ray 1 x Coordinate in meters = -0.267949
Ray 3 x Coordinate in meters = 0.366025
Length in meters = 0.633974
cosine ( AINC) * [ Tangent ( AINC + 15 degrees ) - Tangent ( AINC - 15 degrees ) ] = 0.633974
Length at Surf 90 degrees / Length = 0.845299
Sine ( Surf ) = Cosine ( AINC ) = 0.866025
For SURF 45 degrees and AINC 45 degrees
Ray 1 x Coordinate in meters = - 0.298858
Ray 3 x Coordinate in meters = 0.517638
Length in meters = 0.816496
cosine ( AINC) * [ Tangent ( AINC + 15 degrees ) - Tangent ( AINC - 15 degrees ) ] = 0.816496
Length at Surf 90 degrees / Length = 0.656338
Sine ( Surf ) = Cosine ( AINC ) = 0.707106
For SURF 30 degrees and AINC 60 degrees
Ray 1 x Coordinate in meters = - 0.366025
Ray 3 x Coordinate in meters = 1
Length in meters = 1.366025
cosine ( AINC) * [ Tangent ( AINC + 15 degrees ) - Tangent ( AINC - 15 degrees ) ] = 1.366025
Length at Surf 90 degrees / Length = 0.392304
Sine ( Surf ) = Cosine ( AINC ) = 0.5
The above model gave close results to those based on the standard model of multiplying intensity at 0 degrees angle of incidence by the cosine of Beta to get intensity for Beta degrees angle of incidence.
Coordinate system model 2
Based on a coordinate system in which the coordinates of the light source change and the coordinates of the object hit by the light source stay the same.
Alpha = Angle of incidence for first ray
Beta = Angle of incidence for second ray
Gamma = Angle of incidence for third ray
Delta is in radians
Alpha = Beta - Delta
Gamma = Beta + Delta
Light Source y coordinate = 1 meter * cosine ( Beta )
Light Source x coordinate = 0 meters
The length of the object is 200 meters but the length. width and thickness of the object does not matter for the calculations in this model so long as it is larger than the length the light source hits. A length was arbitrarily assigned so I could give coordinates for the center of the object
Object has x Coordinates from
( - 100 + sine ( Beta ) ) meters to ( 100 + sine ( Beta ) ) meters
Object has y coordinate of 0 for every x coordinate
Center of object hit by light has x coordinate of
1 meter * sine ( Beta )
Distance from light source to center of object = 1 meter
1 meter * cosine ( Beta ) ^ 2 + 1 meter * sine ( Beta ) ^ 2 = 0
When
0 < Delta < Beta
Delta < Beta < 0.5 Pi radians - Delta
Ray 1 hits object at x coordinate of
1 meter * cosine ( Beta ) * tangent ( Beta - Delta )
Ray 2 hits object at exact x coordinate of
1 meter * cosine ( Beta ) * tangent ( Beta ) = 1 meter * sine ( Beta )
Ray 3 hits object at x coordinate of
1 meter * cosine ( Beta ) * tangent ( Beta + Delta)
Length = 1 meter * cosine ( Beta ) * [ tangent ( Beta + Delta ) - tangent ( Beta - Delta ) ]
Template created in geogebra modified to create picture above with image flip
When Beta = 0
Ray 1 hits object at x coordinate of
- 1 meter * tangent ( Delta )
Ray 2 hits object at exact x coordinate of
0 meters
Ray 3 hits object at x coordinate of
1 meter * tangent ( Delta )
Length = 2 * 1 meter * tangent ( Delta )
How to get length equation in detail
Note : The relationship between X and Y and trigonometric functions are "switched" compared with the traditional unit circle because the angle of incident is used instead of the angle relative to the surface. The angle of incident is oriented similar to the other angle at 90 degrees minus theta instead of where the angle theta would be on the unit circle resulting in this "flip flop." This is why I prefer to use the angle relative to the surface instead of the angle of incidence but the angle of incidence is traditionally used in certain physics formulas involving light. In the traditional unit circle x = cosine ( theta ) and y = sine ( theta ) and tangent ( theta ) = slope
Length is the X coordinate value of where ray 3 hits the object minus the x coordinate value of where ray 1 hits the object. This is the "horizontal length used to calculate intensity" in the diagram above
Y = Length B * Cosine ( Beta )
Length B = 1 meter
Y = 1 meter * Cosine ( Beta )
Tangent ( Alpha ) = X value for ray 1 / Y
Tangent ( Beta ) = X value for ray 2 / Y
Tangent ( Gamma ) = X value for ray 3 / Y
X value for ray 1 = Y * Tangent ( Alpha )
Alpha = Beta - Delta
X value for ray 1 = 1 meter * Cosine ( Beta ) * Tangent ( Beta- Delta )
Gamma = Beta + Delta
X value for ray 3 = Y * Tangent ( Gamma )
X value for ray 3 = 1 meter * Cosine ( Beta ) * Tangent ( Beta + Delta )
X value for ray 3 - X value for ray 1 = Length
Length = 1 meter * Cosine ( Beta ) * [ Tangent ( Beta + Delta ) - Tangent ( Beta - Delta ) ]
What are Length G and Length A ?
Y = Length A * Cosine ( Alpha ) = Length B * Cosine ( Beta ) = Length G * Cosine ( Gamma)
Length B = 1 meter
Y = 1 meter * Cosine ( Beta )
Length A = Y / Cosine ( Alpha ) = 1 meter * Cosine ( Beta ) / Cosine ( Alpha )
Length G = Y / Cosine ( Gamma ) = 1 meter * Cosine ( Beta ) / Cosine ( Gamma)
You do not need to use Length G and Length A to calculate intensity
It maybe tempting to try to use length G and length A to calculate intensity using the inverse square law but it does not work like that. Remember the inverse square law is used based on how many rays intersect at a different distance from the source. In the case of the inverse square law the number of rays per surface area is correlated with the length of the rays. Correlation does not always mean causation. The inverse square law is not caused by the length of the rays but the number of rays per surface area. The number of rays in this model is exactly three regardless of the length of each ray therefore in this model length G and length A for rays 1 and 3 are not used to calculate the intensity.
Purpose of providing length G and length A
Length G and Length A are provided in case someone else wants to use those lengths for calculations in some other model or experiment even though they are not important to calculating intensity in this chapter. Possible applications they might matter for include but are not limited to absorption and wavelength and frequency shifting quantities based on the distance of the medium the light ray travels through.
Calculation tables
Rounded truncated to 6 decimal places
Beta = 0 degrees = 0 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
45 * pi /180, 2
40 * pi / 180, 1.678199
35 * pi / 180, 1.400415
30 * pi / 180, 1.154700
25 * pi / 180, 0.932615
20 * pi / 180, 0.727940
15 * pi / 180, 0.535898
10 * pi / 180, 0.352653
5 * pi / 180, 0.174977
4 * pi / 180, 0.139853
3 * pi / 180, 0.104815
2 * pi /180, 0.069841
1 * pi / 180, 0.034910
Delta = 1 degree = 1*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910, 1, 1
5 * pi / 180 radians, 0.035043
10 * pi / 180 radians, 0.0354490
Delta = 2 degree = 2*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
Delta = 3 degree = 3*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
Delta = 4 degree = 4*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
Delta = 5 degree = 5*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
Delta = 10 degree = 10*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
40 * pi / 180 radians,
45 * pi / 180 radians,
50 * pi / 180 radians,
55 * pi / 180 radians,
60 * pi / 180 radians,
65 * pi / 180 radians,
70 * pi / 180 radians,
75 * pi / 180 radians,
80 * pi / 180 radians,
Delta = 15 degree = 15*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
40 * pi / 180 radians,
45 * pi / 180 radians,
50 * pi / 180 radians,
55 * pi / 180 radians,
60 * pi / 180 radians,
65 * pi / 180 radians,
70 * pi / 180 radians,
Delta = 20 degree = 20*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
40 * pi / 180 radians,
45 * pi / 180 radians,
50 * pi / 180 radians,
55 * pi / 180 radians,
60 * pi / 180 radians,
65 * pi / 180 radians,
Delta = 25 degree = 25*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
40 * pi / 180 radians,
45 * pi / 180 radians,
50 * pi / 180 radians,
55 * pi / 180 radians,
60 * pi / 180 radians,
Delta = 30 degree = 30*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
40 * pi / 180 radians,
45 * pi / 180 radians,
50 * pi / 180 radians,
55 * pi / 180 radians,
Delta = 35 degree = 35*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
40 * pi / 180 radians,
45 * pi / 180 radians,
50 * pi / 180 radians,
Delta = 40 degree = 40*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
40 * pi / 180 radians,
45 * pi / 180 radians,
Delta = 45 degree = 45*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
40 * pi / 180 radians,
Delta = 50 degree = 50*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
35 * pi / 180 radians,
Delta = 55 degree = 15*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
30 * pi / 180 radians,
Delta = 60 degree = 60*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
25 * pi / 180 radians,
Delta = 65 degree = 65*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
20 * pi / 180 radians,
Delta = 70 degree = 70*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
15 * pi / 180 radians,
Delta = 75 degree = 75*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
10 * pi / 180 radians,
Delta = 80 degree = 80*pi/180 radians
Beta radians, Length, length / length at 0 radian angle of incident, cosine ( Beta )
0, 0.034910,
5 * pi / 180 radians,
Beta = 5 degrees = 5 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
4 * pi / 180, 0.1403930
3 * pi / 180, 0.105218
2 * pi /180, 0.0701089
1 * pi / 180, 0.035043
Beta = 10 degrees = 10 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
5 * pi / 180, 0.177718
4 * pi / 180, 0.142032
3 * pi / 180, 0.1064415
2 * pi /180, 0.0709216
1 * pi / 180, 0.0354490
Beta = 15 degrees = 15 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
10 * pi / 180, 0.365911
5 * pi / 180, 0.181249
Beta = 20 degrees = 20 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
15 * pi / 180, 0.575767
10 * pi / 180, 0.376838
5 * pi / 180, 0.186395
Beta = 25 degrees = 25 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
20 * pi / 180, 0.827016
15 * pi / 180, 0.600676
10 * pi / 180, 0.391759
5 * pi / 180, 0.193387
Beta = 30 degrees = 30 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
25 * pi / 180, 1.161045
20 * pi / 180, 0.879385
15 * pi / 180, 0.633974
10 * pi / 180, 0.411474
5 * pi / 180, 0.202563
Beta = 35 degrees = 35 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
30 * pi / 180, 1.685010
25 * pi / 180, 1.274374
20 * pi / 180, 0.9503792
15 * pi / 180, 0.678080
10 * pi / 180, 0.437175
5 * pi / 180, 0.214412
Beta = 40 degrees = 40 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
35 * pi / 180, 2.791896
30 * pi / 180, 1.969615
25 * pi / 180, 1.437526
20 * pi / 180, 1.048010
15 * pi / 180, 0.736812
10 * pi / 180, 0.470660
5 * pi / 180, 0.229654
Beta = 45 degrees = 45 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
40 * pi / 180, 8.020403
35 * pi / 180, 3.885519
30 * pi / 180, 2.449489
25 * pi / 180, 1.685394
20 * pi / 180, 1.186666
15 * pi / 180, 0.816496
10 * pi / 180, 0.514731
5 * pi / 180, 0.249364
Beta = 50 degrees = 50 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
35 * pi / 180, 7.174861
30 * pi / 180, 3.411474
25 * pi / 180, 2.099179
20 * pi / 180, 1.394930
15 * pi / 180, 0.928377
10 * pi / 180, 0.573977
5 * pi / 180, 0.275208
Beta = 55 degrees = 55 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
30 * pi / 180, 6.288545
25 * pi / 180, 2.921759
20 * pi / 180, 1.738993
15 * pi / 180, 1.094600
10 * pi / 180, 0.656462
5 * pi / 180, 0.309901
Beta = 60 degrees = 60 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
25 * pi / 180, 5.364922
20 * pi / 180, 2.416091
15 * pi / 180, 1.366025
10 * pi / 180, 0.777861
5 * pi / 180, 0.358179
Beta = 65 degrees = 65 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
20 * pi / 180, 4.407930
15 * pi / 180, 1.893130
10 * pi / 180, 0.973671
5 * pi / 180, 0.429137
Beta = 70 degrees = 70 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
15 * pi / 180, 3.420852
10 * pi / 180, 1.347296
5 * pi / 180, 0.542971
Beta = 75 degrees = 75 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
10 * pi / 180, 2.403275
5 * pi / 180, 0.756736
Beta = 80 degrees = 80 * pi / 180 radians
Delta radians, length, length / length at 0 radian angle of incident, cosine ( Beta )
5 * pi / 180, 1.336743
Disclaimer about my calculations using calculus. I think they are wrong. Which is why I supplied so many tables. If I find or are informed of the mistake I plan to correct it in later editions. If I find out I was correct and why I was correct then I plan to provide that in later editions
The answers I got using calculus look wrong to me based on what I expected to get both based on the angle of incident formula traditionally provided and based on my own calculations using numbers without calculus from tables provided above. I suspect these calculations either are wrong or show that certain formulas should only be used in certain situations. Meaning either the traditional formula for angles of incidence should not be used in the case where my formulas give different results or my formulas based on calculus should not be used when the traditional model or perhaps even my tables give different results.
The way to find out would be to test based on experiments of changing the angle of incidence of a light source at a constant short distance from the center of a flat surface where the rays are not effectively parallel and somehow measuring intensity and or the surface area the light hits at or close to the specific location of the center as opposed to the location of the entire flat surface.
Using calculus and exact coordinates
When
0 < Delta < Beta
Delta < Beta < 0.5 Pi radians - Delta
Ray 1 hits object at x coordinate of
1 meter * cosine ( Beta ) * tangent ( Beta - Delta ) Exact
Ray 2 hits object at exact x coordinate of
1 meter * cosine ( Beta ) * tangent ( Beta ) = 1 meter * sine ( Beta ) Exact
Ray 3 hits object at x coordinate of
1 meter * cosine ( Beta ) * tangent ( Beta + Delta) Exact
Length = 1 meter * cosine ( Beta ) * [ tangent ( Beta + Delta ) - tangent ( Beta - Delta ) ] Exact
If Beta = 0 radians then
Ray 1 hits object at x coordinate of
-1 meter * tangent ( Delta )
Ray 2 hits object at x coordinate of
0
Ray 3 hits object at approximate x coordinate of
1 meter * tangent ( Delta )
Length = 2 meters * tangent ( Delta )
Intensity at angle of incidence Beta / Intensity at angle of incidence at 0 radians equals
Limit as delta approaches 0 radians of
2 * tangent ( Delta ) / ( cosine ( Beta ) * [ tangent ( Beta + Delta ) - tangent ( Beta - Delta ) ] )
is
1 / cos ( Beta)
http://web.archive.org/web/20221229043220/https://www.symbolab.com/solver/limit-substitution-calculator/%5Clim_%7Bx%5Cto%200%7D%5Cleft%282%5Ccdot%5Cfrac%7Btangent%5Cleft%28x%5Cright%29%7D%7Bcos%5Cleft%28k%5Cright%29%5Ccdot%5Cleft%28tangent%5Cleft%28k%2Bx%5Cright%29-tangent%5Cleft%28k-x%5Cright%29%5Cright%29%7D%5Cright%29?or=input
Limit as delta approaches Beta radians from below of
2 * tangent ( Delta ) / ( cosine ( Beta ) * [ tangent ( Beta + Delta ) - tangent ( Beta - Delta ) ] )
is
1 / cos ( Beta)
http://web.archive.org/web/20221230075453/https://www.symbolab.com/solver/limit-substitution-calculator/%5Clim_%7Bx%5Cto%20k%7D%5Cleft%282%5Ccdot%5Cfrac%7Btangent%5Cleft%28x%5Cright%29%7D%7Bcos%5Cleft%28k%5Cright%29%5Ccdot%5Cleft%28tangent%5Cleft%28k%2Bx%5Cright%29-tangent%5Cleft%28k-x%5Cright%29%5Cright%29%7D%5Cright%29?or=input
Note : I was expecting or hoping to get the results of cosine ( Beta) and not it's reciprocal
Using Calculus and Approximate Coordinates
When
0 < Delta < Beta
Delta < Beta < 0.5 Pi radians - Delta
Ray 1 hits object at x coordinate of
1 meter * sine ( Beta - Delta ) Approximate for small values of delta
Ray 2 hits object at exact x coordinate of
1 meter * sine ( Beta ) Approximate for small values of delta
Ray 3 hits object at x coordinate of
1 meter * sine ( Beta + Delta) Approximate for small values of delta
Length = 1 meter * [ sine ( Beta + Delta ) - sine ( Beta - Delta ) ]
When Beta = 0
Ray 1 hits object at x coordinate of
- 1 meter * sine ( Delta ) Approximate for small values of delta
Ray 2 hits object at exact x coordinate of
0 Approximate for small values of delta
Ray 3 hits object at x coordinate of
1 meter * sine ( Delta ) Approximate for small values of delta
Length = 2 meters * sine ( Delta) Approximate for small values of delta
Intensity at angle of incidence Beta / Intensity at angle of incidence at 0 radians equals
Length at angle of incidence at 0 radians / Length at angle of incidence Beta
limit as delta approaches 0 radians of
2 * sine ( Delta ) / ( sine ( Beta + Delta ) - sine ( Beta - Delta ) )
For very small values of delta the sine of delta is approximately equal to delta
limit as delta approaches 0 radians of
2 * Delta / ( sine ( Beta + Delta ) - sine ( Beta - Delta ) ) = 1 / cosine (Beta)
This is the reciprocal of the symmetrical derivative of the sine of Beta with respect to Beta which is equal to the reciprocal of the cosine of Beta
For Intensity at angle of incidence Beta / Intensity at angle of incidence at 0 radians equals
limit as delta approaches Beta radians from below of
2 * sine ( Delta ) / ( sine ( Beta + Delta ) - sine ( Beta - Delta ) )
equals
2 * sine ( Beta ) / ( sine ( 2 * Beta ) ) - sine ( 0 )
equals
2 * sine ( Beta ) / ( 2 * sine ( Beta ) * cos ( Beta ) )
equals
1 / cos ( Beta )
sine ( 2 * theta ) = 2 * sine ( theta ) * cos ( theta )
https://web.archive.org/web/20221211225057/https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle_formulae
Note : I was expecting or hoping to get the results of cosine ( Beta) and not it's reciprocal
Perceived brightness of sun light and size of sun
Angle of incidence
Inverse Square with distance
Decreasing visual size of sun as represented by a angle with decreasing distance
Absorption
Seasons on the heliocentric spinning globe model
Seasons on the globe model are related to the angle of incidence and not primarily the distance the earth is from the sun
The sun is actually closer to the earth not just in summer but also in winter in the heliocentric spinning globe model
Since the sun is closer to the earth during winter than spring and fall but also colder in winter than spring in fall. The angle of incidence is more important than the distance in creating seasonal differences
Seasons on a flat earth model
Seasons on a flat earth model might be related to a change in path the sun floats above the earth
Such changes in path might potentially include a change in altitude above sea level of the sun at different locations during certain seasons
Such changes in path would also potentially mean the sun spends less time at a close enough horizontal distance for daylight and more time far enough away horizontally for it to be night time per day over certain regions during the winter. And the reverse for summer.
Note : If it is summer in the northern hemisphere then it is winter in the southern hemisphere from certain viewpoints. By summer and winter I mean relative to the hemisphere one is referring to.
"For example, the season dates in Australia for 2022 are the following:
Autumn from 1st March 2022 until 31st May 2022
Winter from 1st June 2022 until 31st August 2022
Spring from 1st September 2022 until 30th November 2022
Summer from 1st December 2022 until 28th February 2023"
https://web.archive.org/web/20221027142126/https://takeatumble.com.au/guides/what-months-are-summer-in-australia/
Sunrise and Sunset on the heliocentric globe model
Sunrise and sunset on a globe model are related to the angle of incidence
Sunrise and Sunset on a flat earth model
Sunrise and sunset on a flat earth model would be related to changes in perceived brightness by an observer as the sun floats above the earth in different locations
Changes in perceived brightness would be related to the inverse square law and changes in the distance the sun is from the observer
Changes in perceived brightness would be related to changes in the angle of incidence as the sun changes position relative to the observer
Changes in perceived brightness would be related to changes in the path length the sunlight travels through a medium before reaching a observer as the sun changes position relative to the observer. The greater the path length the more sunlight would be absorbed by such a medium decreasing it's brightness on it's way to the observer.
Sunrise and sunset would also be related to perceived changes in the size of the sun as measured in an angle from the point of view of the observer
Once the sun is to small an angle to be seen and or the brightness from the sunlight is to little to be noticeable by the observer it would be night time in a flat earth model
Sunrise and sunset are related to the sun appearing at a lower angle in the sky if the horizontal distance is farther away when it is at the same vertical altitude above sea level. The sun would not necessarily have to stay at the same altitude above sea level at all times in a flat earth model but if it did there would still be the illusion of sunrise and sunset.
Sun position angle on a two dimensional flat earth model.
It might be better from a technical viewpoint to use a 3D model and a 2D model has certain problems. But a 2D model will be shown because it is simpler. Using a pure 2D model makes calculations related to brightness and incident angle simpler to understand for an hypothetical observer horizontally facing toward the sun. You should never look directly into the sun. This is about a hypothetical observer and I am not suggesting to look at the sun in a real life experiment or in a real life anything.
This is a 2D polar coordinate model with only one angle and one radius to represent the position of the sun relative to an observer. It is combined with a 2D Cartesian coordinate model with one horizontal component and one vertical component.
For these polar coordinates the observer is located at a radius of zero inches away from the origin of the coordinate system.
Pure down is in the direction of the pull of gravity. The flat water surface is perpendicular to the direction of the pull of gravity. Labeling the water surface as flat ignores meniscus and water waves.
For these polar coordinates a 90 degree angle would be perpendicular to the flat surface of water bodies. A positive 90 degree angle would point purely up, and a negative 90 degree angle would point purely down.
This horizontal component of the 2D Cartesian model is parallel to flat surface of water bodies.
This horizontal component of the 2D Cartesian model points directly toward the center of the sun but only in horizontal manner with no vertical component. Positive is horizontally toward the sun and negative is horizontally away from the sun. The direction of the horizontal component of the Cartesian model changes as the sun moves and is different at sunrise and sunset but it is always horizontal.
The vertical component of the 2D Cartesian model is perpendicular to the flat surface of water bodies. Up is positive and down is negative.
The sun might be said to get close to a 90 degree angle at solar noon relative to the observer in this two dimensional model. But for an observer at most locations the sun will never reach 90 degrees even at solar noon. This is because the sun will be at it's shortest horizontal distance from the observer at solar noon but that distance will be greater than zero inches at most locations.
As the sun rises or the sun sets it would become closer and closer to 0 degrees relative to the observer as the sun get's horizontally farther away from the observer. But the sun would never reach 0 degrees because it always has a positive altitude greater than zero inches.
Because the direction of the one and only horizontal Cartesian dimension rotates both sunrise and sunset occur close to 0 degrees instead of one occurring at 0 degrees and the other at 180 degrees in this 2D polar coordinate system
Converting the sun position from cartesian coordinates to polar coordinates
Sun position angle = inverse tangent ( sun altitude / sun horizontal position )
Where the horizontal position is relative to the observer
The sun would appear lower in altitude to the observer as an angle in both sun rise and sun set than at noon in a flat earth model
The sun would disappear as it approaches 0 degrees as the brightness from the light decreases
The sun would disappear as it approaches 0 degrees as the perceived size of the sun decreases when the perceived size is measured as an angle
From a flat earth perspective when the angle approaches 0 degrees then there is an illusion that the sun is going down when it's height does not actually change much and this happens at the same time as the sunlight becomes less bright immediately before night time and people who have a globe earth perspective presume that the sun went down to the other side of the earth which blocks it's light at night time. But according to the flat earth perspective the real cause of the decreased brightness at sunset and night time is that the sun is horizontally farther away from the observer and the sun did not actually go down at all or at least did not go so far down as to be below the observer. By going down I mean it literally is down relative to an observer's local definition of down at night time on a globe earth model.
Solar noon
What Is Noon?
The English language is a little imprecise when it comes to the word “noon”. It can mean 2 different things:
In terms of civil time, noon is simply another word for 12 o'clock, the moment separating morning and afternoon. As such, it is the opposite of midnight. Alternative names include midday and noon time.
In terms of solar time, noon is the moment when the Sun crosses the local meridian and reaches its highest position in the sky, except at the poles. This version of noon is also called solar noon or high noon.
https://web.archive.org/web/20171022111439/https://www.timeanddate.com/astronomy/solar-noon.html
The phrase "highest point" might be problematic on a globe earth model because the sun stays the same distance away from the observer but the earth merely rotates so that the angle changes.
At night time the sun is below the observer on a globe earth model based on their own individual local definition of down. So technically at solar noon the earth is at it's highest point for that observer on a globe earth model based on the observers local definition of up.
The phrase "highest point" is also problematic on a flat earth model because this refers to an angle not a elevation
Sun distance from observer on a flat earth model
The square root of the sun altitude squared plus the sun horizontal distance from the observer squared based on the Pythagorean Theorem
Reflected sun light vs direct hit sun light
There would be two sources of sun light brightness for the observer. Light from the sun hitting the observer directly and light reflecting off on or more object's then hitting the observer. This is before considering the refraction and absorption of light as light travels through mediums.
Angle of incidence and brightness for direct hit sun light on in a flat model
Perceived brightness for a human observer can be measured in lux
If a ray of light hits a surface it will reflect off of or refract with that surface
The angle of incidence is the angle between that ray and a "normal" vector perpendicular to that surface
Accessed online 2022 December 22
Intensity at the surface changes with the Cosine of the angle made with the Normal (perpendicular)
https://www.quora.com/How-are-the-light-angle-and-light-intensity-correlated
The "angle of incidence" is not the same as the "sun position angle"
Sun position angle = inverse tangent ( sun altitude / sun horizontal position )
Angle of incidence = 90 degrees - sun position angle
Brightness = Some quantity * cosine ( Angle of incidence )
cosine ( 90 degrees - Theta ) = sine ( Theta )
cosine ( Angle of incidence ) = sine ( sun position angle )
Brightness = some quantity * sine ( sun position angle )
https://www.quora.com/How-are-the-light-angle-and-light-intensity-correlated
"Some quantity" would not necessarily be a simple constant but would be effected by the factors listed and most likely other factors as well.
Sunlight brightness and the inverse square law on a flat earth model
The brightness from the sun light would decrease as the sun get's farther away from the observer due to the inverse square law
The brightness from the light that hit's the water and ground before reflecting to hit the observer would be more complicated to calculate based on the inverse square law than the simple calculation's for the sun light that hit's the observer directly without reflecting off the ground or water first
Brightness of light hitting observer directly = Some Quantity / Distance of Sun from observer Squared
"Some quantity" would not necessarily be a simple constant but would be effected by the factors listed and most likely other factors as well.
Absorption on a flat earth model
Light traveling through mediums would be absorbed decreasing it's brightness more the farther it travels through a medium in addition to the inverse square effect. When the medium absorbs light it would heat up and or "increase in electron energy levels." But the material of the medium itself would also radiate light as it cools and or "drops in electron energy levels" increasing brightness.
The farther away the sun is from the observer the more sun light brightness would be decreased from absorption on a flat earth model
Clouds would effect absorption in addition to "transparent" air which is not 100% transparent
Refraction sky color and sunrise and sunset color on a flat earth model
Refraction of light traveling through mediums might effect the color the light is perceived to be at different times of day based on the position of the sun relative to the observer on a flat earth model.
This could explain blue skies and different colors of the sky near a sunrise and sunset event location on a flat earth model in a similar way to the explanation of these things due to refraction on a globe earth model.
Sun size on a flat earth model
Perceived size of the sun as an angle in radians = Diameter of the sun / Distance of the sun from the observer
This would not be exact because the sun would be shaped like a line segment with length equal to the diameter of a sphere. But this calculation is for the number of radians in an arc of a section of a circle with a certain perimeter. The length of the line segment is treated as the length of the perimeter of a section of a circle.
Size of the sun on flat earth vs globe earth models
The sun is usually claimed to be a smaller size and closer to the earth in flat earth than globe earth models.
If the sun is N times the distance then it must be N times larger in order to appear the to be the perceived size measured as an angle
I am not arguing for a finite and countable number of photons just because I am explaining a model using photons
I am not saying whether photons actually exist as a countable finite number of particles but modeling light as photon particles helps explain brightness or intensity of light being effected by a inverse square law in the manner that will be described.
People conventionally think of there being a finite and countable number of photons emitted by a source during a time period. However if there were N times as many photons as someone might claim and the amount of brightness each photon contributed to help humans see was divided by N then some but not necessarily all properties of understanding how light works would work the same. If there really were N times as many photons as people believe then instead of each photon having an energy of plank's constant times it's frequency each photon would instead have an energy equal to plank's constant times it's frequency divided by N
standard model with a countable finite number of photons
e = hf
unlimmitted photon model viewing light as a continuous substance instead of a substance made out of countable and finite discrete particles
N times as many photons exist as in standard model and e = hf / N
take the limit as N approaches infinity
The total energy emitted by light sources would be the same because they would have N times as many photons with the energy per photon divided by N
The total brightness by light sources would be the same because they would have N times as many photons with the energy per photon divided by N
I am mentioning this because in another chapter I explain why uncutable atoms might not exist based on the fact that many ( but not necessarily all ) of the physics and chemistry models involving atoms would work to make the same predictions if there were really N times as many atoms in 12 grams of Carbon 12 and each Carbon 12 atom had it's mass divided by N. That is if you did a similar adjustment for all other isotopes of all other elements in relation to the number of atoms in a certain mass and their "atomic mass." You would also have to make certain other adjustments on various quantities measured on a per atom basis.
People might think it is absurd of me to argue against that the general public can know with certainty that there are a finite countable number of atoms in 12 grams of Carbon 12 but support that the general public can know with certainty that there are a finite countable number of photons emitted to produce a certain brightness level, but I assure you that I do not believe that the general public can know with certainty that there are a finite countable number of photons emitted to produce a certain brightness level.
Atomic theory is the scientific theory that matter is composed of particles called atoms. Atomic theory traces its origins to an ancient philosophical tradition known as atomism. According to this idea, if one were to take a lump of matter and cut it into ever smaller pieces, one would eventually reach a point where the pieces could not be further cut into anything smaller. Ancient Greek philosophers called these hypothetical ultimate particles of matter atomos, a word which meant "uncut".
https://web.archive.org/web/20221208012042/https://en.wikipedia.org/wiki/Atomic_theory
The French Catholic priest Pierre Gassendi (1592–1655) revived Epicurean atomism with modifications, arguing that atoms were created by God and, though extremely numerous, are not infinite.
https://web.archive.org/web/20221208012042/https://en.wikipedia.org/wiki/Atomic_theory
Copyright Carl Janssen 2022
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