Point3 Values

Vector Length

length <point3>

Returns the length of the vector.

Vector Dot Product

dot <point3> <point3>

Returns the vector dot product.

The geometric interpretation of the dot product is the length of the projection of the first vector onto the unit vector of the second vector. Obviously, when the two vectors are perpendicular, the dot product is 0.0.

The dot product is commutative, this means that dot X Y == dot Y X.

The dot product is associative, this means that dot (r*X) Y == r*(dot X Y)

The dot product is distributive, this means that dot X (Y+Z) == (dot X Y) + (dot X Z)

Since the dot product of two normal vectors is the cosine of the angle between them, the dot product can be used conveniently to calculate the angle between two vectors:

fn GetVectorsAngle v1 v2 =
(
theAngle = acos(dot (normalize v1) (normalize v2))
)

GetVectorsAngle [10,0,0] [10,20,0]
63.435

Vector Cross Product

cross <point3> <point3>

Returns the vector cross product.

The cross product is a third vector which is always perpendicular to the plane defined by the two vectors, with orientation determined by the right-hand rule.

The cross product can be expressed as the length of the first vector multiplied by the length of the second vector, multiplied by the sine of the angle between the two vectors. Because of this, the cross product of parallel vectors is 0.0 (since sin 0.0 == 0.0)

The cross product is also equal to the Area of the parallelogram formed by the two vectors. Obviously, if the two vectors are edges of a face, the cross product is the normal vector to that face (a vector perpendicular to the face) with length equal to the Area of the face multiplied by 2.

The cross product is anticommutative, this means that cross X Y == cross -Y X

The cross product is associative, this means that cross (r*X) Y == r*(cross X Y)

The cross product is distributive, this means that cross X (Y+Z) == (cross X Y) + (cross X Z)

Normalize Vector

normalize <point3>

Returns the point3 value normalized such that the vector length equals 1.

b =[100,30.5,41.3]--take somePoint3 value
[100,30.5,41.3]
normalB1 =normalize b--getthe normalized vector
[0.889603,0.271329,0.367406]
length normalB1-- check the length, should be 1.0
1.0-- It is!
normalB2 =b / (length b)-- Do-It-Yourself Normalize...
[0.889603,0.271329,0.367406]-- same value
length normalB2-- of course, the length is also 1.0
1.0

Distance Between Two Points / Vectors

distance <point3> <point3>

Returns the distance between the points, equivalent to the length of (point 2 - point 1).

a = [10,20,30]
b = [100,30.5,41.3]
distance a b-- returns 91.3123
distance b a-- returns 91.3123
length (b-a)-- returns 91.3123
length (a-b)-- returns 91.3123 

Pseudo-Random Point / Vector

random <point3> <point3>

Generates a pseudo-random point between the given points. If point A and point B are seen as the corners of a world-oriented bounding box, the random points will NOT reside on the vector B-A but will fill in the volume of the bounding box!

a = [-10,-20,-30]-- define point A
b = [20,30,50]-- define point B
p1 = Point pos:a-- create a Point Helper at point A
p2 = Point pos:b-- create a Point Helper at point B
-- create 2000 spheres at random positions between point A and B
for i = 1 to 2000 do
 sphere pos:(random a b) radius:2 wirecolor:blue

Arbitrary Axis Matrix

arbAxis <point3>

Returns a Matrix3 value representing an arbitrary axis system using point3 as the "up" direction.

Matrix From Normal

matrixFromNormal <point3>

Returns a Matrix3 value with the normal specified by the given point as the Z axis. The translation portion of the Matrix3 value will be [0,0,0]. The components of the scale portion are the inverse of the values required to normalize the point3 value.

See How Do I Get the Local Rotation of a Face? for an example.

MatrixFromNormal [0,0,1]
MatrixFromNormal [0,1,1]

(matrix3 [1,0,0], [0,1,0], [0,0,1], [0,0,0])
(matrix3 [0,-0.707107,0.707107] [1.41421,0,0] [0,1,1] [0,0,0])

3D Noise Functions

noise3 <point3>

A floating point noise function over 3D space, implemented by a pseudo-random tricubic spline. The return value is in the range -1.0 to +1.0. Given the same point3 value, the noise3() function always returns the same value.

noise4 <point3> <phase_float>

The same function as noise3 in which the phase can be varied using the second parameter. The return value is in the range -1.0 to +1.0.

turbulence <point3> <frequency_float>

This is a simple fractal-loop turbulence built on top of the noise3 function. The second parameter controls the frequency of the turbulence. The return value is in the range 0.0 to +1.0.

fractalNoise <point3> <H_float> <lacunarity_float> <octaves_float>

This is a fractal function of 3D space implemented using fractional Brownian motion. The function is homogeneous and isotropic. It returns a float.

The parameters are as follows:

point3 - The point in space at which the noise is computed.

H_float - The fractal increment parameter in the range [0,1]. When H_float is 1 the function is relatively smooth; as H_float goes to 0, the function approaches white noise.

lacunarity_float - The gap between successive frequencies, best set at 2.0.

octaves_float - The number of frequencies in the function.

All the noise functions are based on code and algorithms in the book "Texturing and Modeling: A Procedural Approach", Musgrave, Peachey, Perlin and Worley. Academic Press, ISBN: 0-12-228760-6.

The following script shows the use of various literals, constructors, properties, operators, and methods of the Point3 class.

The following script was written as a test bed for the noise functions. This script displays a 2 dimensional slice of the noise function output as a bitmap. By changing the noise parameters in lines 4 through 10, and specifying which noise function to evaluate in line 12, you can see the effects of changes in the parameters and using different noise functions.

SCRIPT

-- noise functions test bed b_width=320-- specify size of bitmap b_height=320 size=10.-- total distance covered by each row and column z=0.-- z coordinate of 2D slice phase=0.5-- noise4 parameter frequency=10.-- turbulence parameter fract_interval=.5-- fractalNoise parameters lacunarity=2. octaves=5 -- whichfunc=1-- 1 = noise3; 2 = noise4; 3 = turbulence; 4 = fractalNoise -- b=bitmap b_width b_height-- create the bitmap for h=0 to (b_height-1) do-- step through each row of the bitmap ( h_norm=(h as float/(b_height-1))*size-- calculate y coordinate row = for w=0 to (b_width-1) collect-- collect row of pixel colors ( w_norm=(w as float/(b_width-1))*size-- calculate x coordinate noise_val = case whichfunc of-- store result of selected function ( 1: noise3 [w_norm, h_norm , z] 2: noise4 [w_norm, h_norm , z] phase 3: turbulence [w_norm, h_norm , z] frequency 4: fractalNoise [w_norm, h_norm , z] fract_interval lacunarity octaves ) noise_val = 0.5*(1.+noise_val)-- convert output range to 0. to 1. white*noise_val-- and multiply by color white ) setpixels b [0,h] row-- store row of pixels in bitmap ) display b-- display bitmap in VFB

OUTPUT

The following figures show the bitmap generated by the above script for each noise function type.

noise3 noise4 turbulence fractalNoise