Programming Math Notes

circle_and_arcs.txt

Radius = ArcLength / Angle
ArcLength = Radius * Angle
Arc / Segment Height = Radius * ( 1 - cos(angle / 2) )
circleCenterPoint_fromCurve = Radius - Segment Height

chord length = 2 * Radius * sin( angle / 2 )

circumference = PI * diameter
circumference = PI * radius * 2;

arcLen / circumference = degree ratio (Meaing 10 degress over 360 degrees)

derivatives.txt

https://www.mathsisfun.com/calculus/derivatives-rules.html

Sum / Difference (+ or -) : treat each side as a function that you need to make a derivative of.
f + g -> f' + g'

constant = 0, variable = 1 :: x - 1 become 1 - 0

const * var :  D(2x)  ->  2 D(x)  ->  2 * 1

sqrt(x) : 1 / (2 * sqrt(x))

function * variable = ax -> a
	(-P0 + P2) * t -> (-P0 + P2)

Power Rule :  x^2 = D( n*x^n-1 ) = 2x^1
			 example2:	3x^2  ->  3 * 2x^1  ->  6x^1

Product Rule : f() * g() =  f() * g'() + g() * f'();
	fg -> f g' + f' g

Chain Rule :  f(g()) = f'( g() ) * g'();

Quatient Rule : f() / g() :: g()*f'() - f() * g'() / g() * g()
	f/g ->  (f' g - g' f) / g^2

Reciprocal Rule
	1 / f -> -f' / f^2

inversCOT(x) = -1 / 1 + x^2

sin(x) = cos(x)
cos(x) = -sin(x)
tan(x) = sec^2( x )

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EXAMPLES
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(2t³ - 3t² + 1) * p0 + 
(t³ - 2t² + t) * m0 + 
(-2t³ + 3t²) * p1 + 
(t³ - t²) * m1


(6t² - 6t)p0 + 
(3t² - 4t + 1)m0 + 
(-6t² + 6t)p1 + 
(3t² - 2t)m1


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0.5 * ( 
    (2 * P1) +
    (-P0 + P2) * t + 
    (2*P0 - 5*P1 + 4*P2 - P3) * t² + 
    (-P0 + 3*P1- 3*P2 + P3) * t³
 )


2 * P1 -> 0 * 1 = 0
(-P0 + P2) * t -> ax -> a --> (-P0 + P2 )

(2*P0 - 5*P1 + 4*P2 - P3) * t²
	(2*P0 - 5*P1 + 4*P2 - P3) * 2t
	(-P0 + 3*P1- 3*P2 + P3) * t³


0.5 *( 
	(-P0 + P2) + 
	2*(2*P0 - 5*P1 + 4*P2 - P3) * t + 
	3*(-P0 + 3*P1- 3*P2 + P3) * t²
)


----------------------------------------------
(s*P0 - s*P1) + 
( (s - 2) * P0 + (3-2*s) * P1 + s*P2 ) * t +
( (2-z) * P0 + P3  ) * t^2 +
( -s * P0 + 2*s*P1-s * P2 ) * t^3

( (s-2) *P0 + (3-2*s)*P1 + s*P2 ) +
2 * ((2-s)*P0 +                    P3  ) * t
3 * (-s*P0    + 2*s*P1      -s*P2      ) * t^2


p1.x + d0x * t + (- 3 * p1.x + 3 * p2.x - 2 * d0x - d1x) * tt + ( 2 * p1.x - 2 * p2.x + d0x + d1x) * ttt
d0x + 2 * (- 3 * p1.x + 3 * p2.x - 2 * d0x - d1x) * t + 3 * ( 2 * p1.x - 2 * p2.x + d0x + d1x) * tt

kinematic_suvat_equations.txt

https://www.youtube.com/watch?v=v1V3T5BPd7E
https://www.youtube.com/watch?v=phMZQNu0ZFM
https://www.youtube.com/watch?v=IvT8hjy6q4o

s = displacement (distance)
u = inital velocity
v = final velocity
a = acceleration
t = total time

1. s = (( u + v ) / 2) * t
	Average Velocity = displacement / time 
	(inital v + final v) / 2 = displayment / time

2. v = u + a * t
	a = (v - u) / t
	t = (v - u) / a
	Constant Accel = (change of velocity) / time
	Constant Accel = (final vel - initial vel) / time

3. s = (u * t) + (( a * t^2 ) / 2)

4. s = v * t - ((a * t^2) / 2 )

5. v^2 = u^2 + 2 * a * s
	u^2 = v^2 - 2 * a * s

6. s = (v^2 - u^2) / (2 * a)

law_cos_sin.txt

(lowercase is length, Upper is Angle)
    C
   /\ 
b /  \ a
A/____\ B
   c
longer side is C

	*when knowing 2 angles, subtract both from 180 to get remaining angle.

	SSS : Solve only knowing sides Side-Side-Side  
	cos(C) = (a^2 + b^2 - c^2) / 2ab
	cos(A) = (b^2 + c^2 - a^2) / 2bc
	cos(B) = (c^2 + a^2 - b^2) / 2ca


	SSA : Solve length if 2 sides and its angle is known
	c^2 = a^2 + b^2 - 2ab cos(C)
	a^2 = b^2 + c^2 - 2bc cos(A)
	b^2 = c^2 + a^2 - 2ca cos(B)

Law of Sines
	
	SSA : find Angle of C when knowning Length of A and B, angle of B.
	sin(C) = (a * Math.sin(B)) / b   

misc.txt

abs( fract( X - 0.5 ) - 0.5 )
	Zero will Equal 1
	Any Whole Number Will Equal 0
	Any 0.5 number will Equal 0.5
	Any Number over before or after 0.5 will be its opposite on the 0-1 range.
	0 = 1 -> 0.2 = 0.8 -> 0.5 = 0.5 -> 0.7 = 0.3 -> 1.0 = 0
	
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Javascript :  1.6 | 0 = 1.0  // Using Pipe works like Math.floor

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Instead of using x % 2 = 0 or 1
Use bitwise operations  x & 1 == 1
	if the value is even, will equal 0
	if the value is odd, will equal 1

projection.txt

Closet Point on a plane from a point.
    p = point
	v = p - planePoint  //vector length from point to a plane's point.
	n = the plane's unit length vector direction
	len = v dot n
	pos = p - (n * len)

Intersection Point between a Plane and a Ray

	dot( planePos-rayOrigin, planeNorm ) / dot(rayVecLen, planeNorm);