CSCI 431 Ch. 4 Terminology

The origin and the basis vectors determine a _______.

frame

rule used to combine directed line segments

head to tail rule

Use a four-dimensional representation for both points and vectors in three dimensions

homogeneous coordinates

does a vector have a fixed location in space?

no

The vector n is perpendicular, or orthogonal, to the plane; it is called the _______ to the plane.

normal

Usually the natural frame of the application

object or world frame

If u ⋅ v = 0, u and v are said to be ________.

orthogonal

The forms P(α), for the line, and T(α,β), for the plane, are known as _______ forms because they give the value of a point in space for each value of the parameters α and β.

parametric

A ________ in an affine space can be defined as a direct extension of the parametric line.

plane

in a three-dimensional geometric system, it is a location in space

point

simplest geometric object

point

uppercase letters P, Q, R, . . . denote ________ scalars points vectors

points

Negative values of α give us ______ about the fixed point, in the scaling direction.

reflection

Rotation and translation are known as _______ transformations. No combination of rotations and translations can alter the shape or volume of an object; they can alter only the object's location and orientation.

rigid-body

lowercase letters u, v, w, . . . denote ________ scalars points vectors

vectors

A vector space contains two distinct types of entities:

vectors and scalars

special vector, has a magnitude of zero, has no length, orientation is undefined

zero vector

The following is the usual order in which the frames occur in the pipeline

1. model coordinates 2. object or world coordinates 3. eye or camera coordinates 4. clip coordinates 5. normalized device coordinates 6. window or screen coordinates

An _______ is a set of operations on data; the operations are defined independently of how the data are represented internally or of how the operations are implemented.

ADT

A ________ is an extension of a vector space that adds a measure of size or distance and allows us to define such things as the magnitude of a vector.

Euclidean space

affine addition

P = Q + (alpha)v

point vector addition

P = Q + v

parametric form

P(alpha) = P_0 +(alpha)d

To specify a translation, we need only specify a displacement vector d, because the transformed points are given by ______ for all points P on the object.

P′ = P + d

Three features characterize three-dimensional objects that fit well with existing graphics hardware and software:

The objects are described by their surfaces and can be thought of as being hollow. The objects can be specified through a set of vertices in three dimensions. The objects either are composed of or can be approximated by triangles.

adds vectors 'b' and 'c' and puts result in 'a'

a = add(b, c);

Although the mathematician may prefer to think of scalars, points, and vectors as members of sets that can be combined according to certain axioms, the computer scientist prefers to see them as ________.

abstract data types (ADTs)

An _______ is an extension of the vector space that includes an additional type of object: the point.

affine space

In an n-dimensional space, a _____ is a set of n linearly independent vectors. Given a ______ v1,v2,...,vn, any vector in the space can be expressed uniquely as v = α1v1 + α2v2 +...+ αnvn.

basis

At some point, we want to know how these objects appear to the camera. It is natural at that point to convert representations from the world frame to the _______ frame.

camera or eye

Is WebGL based on row major order or column major order?

column major order

In a three-dimensional vector space, we can represent any vector v uniquely in terms of any three linearly independent vectors v1,v2, andv3 as v=α1v1+α2v2+α3v3. The scalars α1,α2, and α3 are the ______ of v with respect to the basis v1,v2, and v3.

components

In such systems, we build objects from a small set of volumetric objects through a set of operations such as union and intersection.

constructive solid geometry (CSG)

A ________ object is one for which any point lying on the line segment connecting any two points belonging to the object also belongs to the object.

convex

Geometrically, the _________ is the set of points that we form by stretching a tight-fitting surface over the given set of points—shrink-wrapping the points. It is the smallest convex object that includes the set of points.

convex hull

The set of points formed by the affine sum of n points, under the additional restriction α_i ≥ 0, i=1,2,...,n, is called the _________ of the set of points

convex hull

We can also use two nonparallel vectors, u and v, to determine a third vector n that is orthogonal to them. This vector is the _______ : n = u × v.

cross product

sets 'd' to an identity matrix

d = mat4();

sets 'd' to the transpose of 'e'

d = transpose(e);

A vector space has a ________, which is the maximum number of linearly independent vectors that we can find.

dimension

The ______ of u and v is written u · v

dot product

sets 'f' to the product of 'e' and 'd'

f = mult(e, d);

For a Euclidean space of line segments, the magnitude of a segment is its _______.

length

In a vector space, a set of vectors is _______ if we cannot write one of the vectors in terms of the others using scalar-vector multiplication and vector addition.

linearly independent

which properties does a point possess: size shape location

location

The _________ of a vector v is a real number denoted by |v|.

magnitude

in WebGL, we specify our geometry using the coordinate system or frame that is natural for the model, which is known as the ________

model frame

method of specifying a matrix m; elements of the first row, followed by the elements of the second row, and so on.

row major order

We start with a set of scalars, any pair of which can be combined to form another scalar through two operations, called addition and multiplication. If these operations obey the closure, associativity, commutativity, and inverse properties described in Appendix B, the elements form a _________.

scalar field

we can combine scalars and vectors to form new vectors through ________

scalar-vector multiplication

Greek letters α, β , γ , . . . denote _______ scalars points vectors

scalars

objects that obey a set of rules that are abstractions of the operations of ordinary arithmetic

scalars

_______ is an affine non-rigid-body transformation by which we can make an object bigger or smaller.

scaling

Because triangular polygons are always flat, either the modeling system is designed to always produce triangles, or the graphics system provides a method to divide, or _________, an arbitrary polygon into triangular polygons.

tessellate

the most important mathematical space

the linear vector space

A _______ is a function that takes a point (or vector) and maps it into another point (or vector).

transformation

_________ is an operation that displaces points by a fixed distance in a given direction

translation

Hence, even if our modeling system provides curved objects, we assume that a _______ mesh approximation is used for rendering.

triangle

point point subtraction

v = P - Q

creates a vec3 with all components set to 0

var a = vec3();

creates a vec3 with the components 1, 2, 3

var b = vec3(1, 2, 3);

copies the vec3 'c' by copying vec3 'b'

var c = vec3(b);

creates a mat3 identity matrix

var d = mat3();

creates a mat3 from 9 elements

var e = mat3( 0, 1, 2, 3, 4, 5, 6, 7, 8);

creates the mat3 'f' by copying mat3 'e'

var f = mat3(e);

any quantity with direction and magnitude

vector

we can combine vectors with vectors through _______

vector-vector addition