Table of Contents
There are three kinds of lies: lies, damned lies, and statistics. – Benjamin Disraeli (1804-1881), British politician
THEORY
[[Software]]
http://www.math.com/students/calculators/source/scientific.htm WebService ?
Mathematica http://www.wolfram.com/
Maple http://www.maplesoft.com/
GNU MP http://www.swox.com/gmp/
Scilab http://www.scilab.org/
MathLab mathworks ?
[[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
