3D Vector arithmetic and THREE.Vector3

CS 493 Lecture, Dr. Lawlor

Vectors, dot, and cross products (warp-speed review)

You should know basic 2D and 3D vectors, including dot products and cross products. Vectors are basically all we ever deal with in computer graphics, so it's important to understand these pretty well.

In particular, you should know at least:

The difference between a 3D vector, like "vec3"; and a 1D scalar, like "float". I try to consistently write vectors in uppercase, like "vec3 A", and scalars in lowercase, like "float a". You can't add vectors and scalars, so you do need to keep track of the difference.
Vectors as positions/locations/coordinates ("absolute", with w==1.0), shifts/offsets/directions ("relative", with w==0.0), and unit-length directions ("normalized", a special case of relative vectors).

Know how to calculate a vector's length, via the Euclidian distance: len=sqrt(x*x+y*y+z*z);
Know that "to normalize" means to make a vector have unit length (for example, by dividing it by its length).

Homogenous vector interpretation: (x,y,z,1) is an absolute location; (x,y,z,0) is a relative direction.
"It's just a bag of floats" vector interpretation: the color (r,g,b,a) can and is treated like as a vector, especially on the graphics hardware.
Dot product: compute the "similarity" of two vectors

A dot B = A.x*B.x + A.y*B.y + A.z*B.z

For unit vectors, A dot B is the cosine of the angle between A and B.

You can figure this out if you write down the dot product of two unit 2D vectors: A=(1,0) lying along the x axis, B=(cos theta, sin theta) in polar coordinates. Then A dot B = cos theta.

For unit vectors, A dot B is:

1 if A and B lie in the same direction (angle=0)
0 if A and B are orthogonal (angle=90)
-1 if A and B lie in the opposite directions (angle=180)
Always the "signed length" of the projection of A onto B (or vice versa).

For any vector, A dot A is the square of A's length. For a unit vector, A dot A is 1.
In general, A dot B = length(A) * length(B) * cos(angle between A and B)
Dot product is commutative: A dot B = B dot A
Dot product distributes like multiplication: A dot (B + C) = A dot B + A dot C
Dot product is friendly with scalars: c*(A dot B) = (c*A) dot B = A dot (c*B)
"A dot B" is the math notation. In code you usually write "float f = dot(A,B);" or "var f=A.dot(B);".
Dot product is very handy in computer graphics for computing diffuse (Lambertian) lighting: for surface normal N and light direction L, "float light = dot(N,L);".

Coordinate frames

A set of 3 orthogonal and normalized (orthonormal) vectors (call them S, U, and V) is actually a rotation matrix, and it has some magical properties.
The magic is if I take any point X=s*S+u*U+v*V, then

X dot S = (s*S+u*U+v*V) dot S
X dot S = s * (S dot S) + u * (U dot S) + v * (V dot S)
X dot S = s*1 + u * 0 + v * 0 (because S is unit, and S, U, and V are orthogonal)
or X dot S = s

That is, the 's' coordinate of the (S,U,V) coordinate system is just X dot S!

To get into the coordinate system: dot the incoming vector with each axis.

s=X dot S; u=X dot U; v=X dot V;

To get out of the coordinate system: scale each axis by the coordinates.

X=s*S+u*U+v*V;

To put the coordinate origin at W, just subtract W from X before dotting (or, subtract W dot S after dotting)

Cross product: compute a new vector orthogonal to the two source vectors

A cross B = (A.y*B.z-A.z*B.y, A.z*B.x-A.x*B.z, A.x*B.y-A.y*B.x)

On your right hand, if your index finger is A, and your middle finger is B, your thumb points along "A cross B".
"A cross B" is the math notation. In code you usually write "vec3 C = cross(A,B);" or "var C=A.cross(B);".
The vector A cross B is orthogonal to both A and B. (dot(A,C) = dot(B,C) = 0)
length(cross(A,B)) == length(A)*length(B)*sin(angle between A and B)
Cross product can be used to make any two non-orthogonal vectors A and B into an orthogonal frame (X,Y,Z). The idiomatic use is:

X = A (pick a vector to be one axis)
Z = A cross B (new 'Z' axis is orthogonal to A/B plane)
Y = Z cross X

Note that the cross product is anti-commutative: A cross B = - B cross A

JavaScript and THREE.Vector3

There are several annoying things about doing 3D vector work in JavaScript:

No operator overloading, so you need "a.addSelf(b)" instead of "a+=b". This makes expressions like "P+=dt*V+0.5*dt*dt*A;" quite painful.
No function overloading, so "dot(a,b)" or "normalize(c)" can't work with both vec2 and vec3. Instead, it's a method call like "a.dot(b)" or "c.normalize()".
Low performance, even for a CPU.

One of the most frustrating things about doing high performance code in interpreted languages is the high cost of dynamic allocations with "new". C++ makes extensive use of stack allocation, so "vec3 a=2.0;" gets inlined and transformed into registers or temporaries by the compiler. By contrast, in interpreted languages, this is a memory allocation. Thus the API contorts to avoid creating new objects, typically by providing interfaces that mostly mutate existing objects rather than creating new objects. You can work around this by explicitly asking for new objects (e.g., ".clone()") when you'd prefer not to trash the old values.

For example:

Operation	GLSL or "osl/vec3.h"	THREE.Vector3;
Vector addition	vec3 c=a+b;	var c=a.clone().addSelf(b);
Incremental addition	c+=d;	c.addSelf(d);
Vector scaling	vec3 s=a*0.5;	var s=a.clone().multiplyScalar(0.5);
Dot product	float x=dot(a,b);	var x=a.dot(b);
Cosine of angle between vectors	float cos_ang=dot(a,b) / (length(a)*length(b));	var cos_ang=a.dot(b) / (a.length()*b.length());
Make unit vector	vec3 d=normalize(p);	var d=p.clone().normalize();

For examples, see the calculations in the boundary planes demo or the boundary sphere demo.