There are three kinds of lies: lies, damned lies, and statistics. – Benjamin Disraeli (1804-1881), British politician

THEORY

[[Software]]

[[Programming]]

TRIGONOMETRY

sin = Op / Hyp
cos = Adj / Hyp
tan = Op / Adj

Quaternions @ http://en.wikipedia.org/wiki/Quaternion

http://skal.planet-d.net/demo/matrixfaq.htm

Trigo : http://science.slashdot.org/article.pl?sid=05/09/17/1313249&from=rss

Javahttp://mrl.nyu.edu/~perlin/courses/fall97/improv-docs/improv.util.Quaternion.html

Rotation :

  • Vector + angle { x y z a }
  • Quaterion : { sin(a/2)* x , sin(a/2) * y , sin(a/2) * z , cos(a/2) }

LINEAR ALEBRA

norm :

|AB| = sqrt( X^2 + Y^2 + Z^2 )

dot product (scalar) with a '.' :

A . B = |A| * |B| * cos(A,B)
      = Ax*Bx + Ay*By + Az*Bz

props:

X . Y = Y . X  # commutative
( a * X ) . Y  =  a * ( X . Y ) # associative
X . ( Y + Z ) = ( X . Y ) + ( Y . Z ) # distributive

cross product (vect) with a 'x' : (result is called normal)

A x B = ( Ay*Bz-By*Az , Az*Bx-Bz*Ax , Ax*By-Bx*Ay )
| A x B | = |A| * |B| * sin(a,b)  (which is the area of the paralellogram)

props:

A x A = 0
A x B = -(B x A),
r being a constant, (r * A) x B = r * ( A x B )
A x (B + C) = A x B + A x C,
( A x B = C ) and respectively ( C x A = B ) and ( B x C = A )

APPLICATIONS

angle (A,B) :

acos ( A.B / |A|*|B| )
asin ( |AxB| / |A|*|B| )

normal:

A x B is the normal of the 2 vectors (2 rights angles)
order must be be anticlockwise for a (right hand direction)

projection: (of unit vectors)

A'  = A projected on B = (A . B) * B

distance:

within a Point P and a Unit Vector D staring in O :
d = D x ( P - O  ) # check it
 = sqrt (
 ( Dy * ( Pz - Oz ) - Dz * ( Py - Oy ) )^2
+ ( Dz * ( Px - Ox ) - Dx * ( Pz - Oz ) )^2
+ ( Dx * ( Py - Oy ) - Dy x ( Px - Ox ) )^2
)
if we are in a 2D world :
d = abs ( Dx * ( Py - Oy ) - Dy x ( Px - Ox ) )

OBJECTS

line:

P = O + l * D
# l is the parametric factor & O origin
# D may be any vector including unit one or (M-O), M is anywhere on line

plane:

define it by ( O & N  ) or ( N & k )
N . P  = k # k is constant and N normal of plane
( Nx * Px ) + ( Ny * Py ) + ( Nz * Pz ) - k = 0

sphere :

| P - C | = r # C is Center and r radius

distance point to plane : ( O is a Point of Plane)

d = ( ( N . P ) - ( O . N ) ) / norm(N) # check it
 = N . ( P - O ) / norm(N)  # check it

intesection : line & plane

[[ P.N = k
[[ P = O + l * D
l = ( k - ( O . N ) ) /  ( D . N )

intersection : plane & plane & plane

[[ P.N1 = k1
[[ P.N2 = k2
[[ P.N3 = k3
TODO, see matrix & inv mtx

intersection : plane & plane

[[ P.N1 = k1
[[ P.N2 = k2

intersection : line & sphere

[[ P = O + l * D
[[ | P - C | = r
TODO , see quadratic

ROTATIONS MATRIX

around X :

x' =   x
y' =   (cos a * y) - (sin a * z)
z' =   (sin a * y) + (cos a * z)

around Y :

x' =   (cos a * x) + (sin a * z)
y' =   y
z' = - (sin a * x) + (cos a * z)

around Z :

x' =   (cos a * x) - (sin a * y)
y' =   (sin a * x) + (cos a * y)
z' =   z

MATHS

question : comment determiner (n) a partir de (x) sachant que :

2^(n-1) < x << 2^n ?
(n-1) log 2 < log x << n log 2
(n-1) < log x/log 2 << n
double n = log ( x  ) * 1. / log (2.) ;
if ( n - ( (double) (int) n) ) n++;
int m = 1<<(int) n;

Another solution is to use bitmask :

1/ o = count '1' bits
2/ if o = 1 return o's bit position
3/ else shift right
4/ return 1st '1' bit'position from left

MISC

MORE

maths.txt · Last modified: 2018/05/21 11:29 by rzr
 
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