Geometries

Question 1

Vector Spaces

Answer 1

Set of vectors with addition and scalar multiplication defined.

Question 2

Axioms of vector spaces

Answer 2

1. Addition commutes
2. Addition associates
3. Scalar Multiplication distributes
4. Unique Zero Element
5. Field unit element

Question 3

Span

Answer 3

Set B spans V iff any vector in V can be written as a linear combination of vectors in B.

Question 4

Basis

Answer 4

A minimal spanning set (all bases are the same size)

Question 5

Dimension

Answer 5

Number of vectors in a basis

Question 6

Affine Space

Answer 6

A set of vectors V and points P such that p + v is also in P.

Question 7

Frame

Answer 7

An affine extension of a basis. A set of vectors and an origin point.

Question 8

Dimension of an affine space

Answer 8

The same as the dimension of a vector space

Question 9

Inner Product Space

Answer 9

Inner product is a binary operator for any vector space V with:

1. Transitivity
2. Commutativity
3. v dot v >= 0

Question 10

Euclidean Spaces

Answer 10

A space with a distance metric defined by the inner product .

Question 11

Perpendicularity

Answer 11

u dot v = 0 implies u and v are perpendicular.

Not an affine concept. There are no angles in affine space.

Question 12

Metric Space

Answer 12

Any space with a distance metric d(P, Q) defined on its elements.

Question 13

Distance Metric

Answer 13

Must satisfy the following axioms:

1. d(P, Q) >= 0
2. d(P, Q) = 0 iff P=Q
3. d(P, Q) = d(Q, P)
4. d(P, Q) <= d(P, R) + d(R, Q)

Distance is intrinsic to the space and not a property of the frame.

Question 14

How to find the angle between two vectors

Answer 14

cosinv(u dot v / (norm(u) * norm(v)))

Question 15

Cartesian Space

Answer 15

A Euclidean space with a standard Orthonormal Frame

Question 16

Orthonormal frame

Answer 16

A frame where all the vectors are perpendicular and normalized so that their magnitude = 1.

Question 17

Cartesian Coordinates

Answer 17

Vectors: 0
Points: 1

Question 18

The problem with vector spaces

Answer 18

There is no distance metric and transforming objects is impossible since vectors can’t translate.

Question 19

Affine combinations

Answer 19

1. Point Subtraction

2. Point Blending A + t(B-A) = C

Question 20

Parametric Line Equation

Answer 20

L(t) = A + t(B - A) = (1-t)A + tB

Question 21

Parametric Ray Equation

Answer 21

R(t) = A + t d

Where A is the origin and d is the direction of the ray

Question 22

Affine transformations

Answer 22

A mapping T from affine space A1 to A2 is an affine transformation if

1. T maps vectors to vectors and points to points
2. T is a linear transformation on the vectors
3. T(P + u) = T(P) + T(u)

T also preserves affine combinations on the points.

Question 23

Which is a larger class of transformations: Affine or Linear?

Answer 23

Affine transformations are a bigger class.

Question 24

What happens to parallel lines that have an affine transformation applied to them?

Answer 24

They remain parallel

Question 25

4 Parameters for specifying a transformation

Answer 25

1. A matrix
2. Domain Space
3. Range Space
4. Coordinate frame in each space