Mathematical Foundations

Sections

A-1 to A-5

Some of the material in these slides may have been adapted from university of Virginia, MIT and Åbo Akademi University

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Mathematical Foundations

● I’ll give a brief, informal review of some of the mathematical tools we’ll employ■ Geometry (2D, 3D)■ Trigonometry■ Vector and affine spaces

○ Points, vectors, and coordinates

■ Dot and cross products■ Linear transforms and matrices

● I know that everyone knows that…

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2D Geometry

● Know your high-school geometry:■ Total angle around a circle is 360° or 2π radians■ When two lines cross:

○ Opposite angles are equivalent○ Angles along line sum to 180°

■ Similar triangles:○ All corresponding angles are equivalent○ Corresponding pairs of sides have the same length ratio and are

separated by equivalent angles○ Any corresponding pairs of sides have same length ratio

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Trigonometry

● Sine: “opposite over hypotenuse”● Cosine: “adjacent over hypotenuse”● Tangent: “opposite over adjacent”● Unit circle definitions:

■ sin () = y/z ■ cos () = x/z■ tan () = y/x■ Z = sqrt(x2 + y2)■ Degrees vs Radians■ Etc…

y

x

Z

5

3D Geometry

● To model, animate, and render 3D scenes, we must specify:■ Location■ Displacement from arbitrary locations■ Orientation

● We’ll look at two types of spaces:■ Vector spaces■ Affine spaces

● We will often be sloppy about the distinction

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Co-ordinates

● Cartesian■ 3 Perpendicular Axes■ Right Hand Rule

● Curvilinear■ Any non-cartesian■ Formed by the intersection of any 3 surfaces■ Each Surface is constructed by fixinging one co-ordinate■ May be orthogonal

y

x

z

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Cylindrical Co-ordinate

z

• Related to Cartesian• A surface is formed by fixing one co-ordinate• Plane including the z-axis is formed by fixing • Plane intersecting z-axis is formed by fixing z• Cylinder is formed by fixing

x-axis

y-axis

● How to convert from Cylindrical to Cartesian

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Polar Co-ordinates

• 3 co-ordinates: • Surface of constant is sphere• Surface of constant is a cone• Surface of a constant is a plane including z- axis

z

x-axis

y-axis

● How to convert from Polar to Cartesian

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Vector Spaces

● A space is basically a set of elements with operations defined on the elements

● Two types of elements:■ Scalars (real numbers): … ■ Vectors (n-tuples): u, v, w, …

● Supports two operations:■ Addition operation u + v, with:

○ Identity 0 v + 0 = v○ Inverse - v + (-v) = 0

■ Scalar multiplication:○ Distributive rule: (u + v) = (u) + (v)

( + )u = u + u

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Vector Spaces

● A linear combination of vectors results in a new vector:

v = 1v1 + 2v2 + … + nvn

● If the only set of scalars such that

1v1 + 2v2 + … + nvn = 0

is 1 = 2 = … = 3 = 0

then we say the vectors are linearly independent● The dimension of a space is the greatest number of linearly

independent vectors possible in a vector set● For a vector space of dimension n, any set of n linearly

independent vectors form a basis

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Vector Spaces: A Familiar Example

● Our common notion of vectors in a 2D plane is (you guessed it) a vector space:

■ Vectors are “arrows” rooted at the origin■ Scalar multiplication “streches” the arrow, changing its length

(magnitude) but not its direction■ Addition can be constructed using parallelogram ( “trapezoid rule”)

u+vy

xu

v

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Vector Spaces: Basis Vectors

● Given a basis for a vector space:■ Each vector in the space is a unique linear combination of the basis

vectors■ The coordinates of a vector are the scalars from this linear combination■ Orthonormal Basis

○ Orthogonal uk.uj = 0, j k

○ Unit length uk.uk = 1

■ Best-known example: Cartesian coordinates○ Draw example on the board○ (i, j, k)

■ Note that a given vector v will have different coordinates for different bases

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Vectors And Points

● We commonly use vectors to represent:■ Direction (i.e., orientation)■ Points in space (i.e., location)■ Displacements from point to point

● But we want points and directions to behave differently■ Ex: To translate something means to move it without

changing its orientation■ Translation of a point = different point■ Translation of a direction = same direction

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Affine Spaces

● An affine space is what is left of a vector space after you've forgotten which point is the origin.

● Points support these operations■ Point-point subtraction: Q - P = v

○ Result is a vector pointing from P to Q

■ Vector-point addition: P + v = Q○ Result is a new point○ P + 0 = P

■ Note that the addition of two points is undefined■ Scaler times point is undefined

P

Q

v

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Affine Spaces

● Points, like vectors, can be expressed in coordinates■ The definition uses an affine combination■ Net effect is same: expressing a point in terms of a basis

● Thus the common practice of representing points as vectors with coordinates

● Be careful to avoid nonsensical operations■ Point + point■ Scalar * point

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Affine Lines: An Aside

● Parametric representation of a line with a direction vector d and a point P1 on the line:

P() = Porigin + d

● Restricting 0 produces a ray in the direction of the vector d

● Setting d to P - Q and restricting 1 0 produces a line segment between P and Q

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Dot Product

● The dot product or, more generally, inner product of two vectors is a scalar:

v1 • v2 = x1x2 + y1y2 + z1z2 (in 3D)

● Useful for many purposes■ Computing the length of a vector: length(v) = sqrt(v • v)

■ Normalizing a vector, making it unit-length■ Computing the angle between two vectors:

u • v = |u| |v| cos(θ)

■ Checking two vectors for orthogonality■ Projecting one vector onto another

θu

v

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Cross Product

● The cross product or vector product of two vectors is a vector:

● v1xv2 = u|v1||v2|sin ● Cross product of two vectors is orthogonal to both● Right-hand rule dictates direction of cross product● Cross product is handy for finding surface orientation

■ Lighting■ Visibility

● Not Associative● Distributive

1221

1221

1221

21 )(

y x y x

z x z x

z y z y

vv

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Linear Transformations

● A linear transformation: ■ Maps one vector to another■ Preserves linear combinations

● Thus behavior of linear transformation is completely determined by what it does to a basis

● Turns out any linear transform can be represented by a matrix

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Matrices

● By convention, matrix element Mrc is located at row r and column c:

● By convention (mathematical) , vectors are columns:

mnm2m1

2n2221

1n1211

MMM

MMM

MMM

M

3

2

v

v

v

v

1

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Matrices

● Matrix-vector multiplication applies a linear transformation to a vector:

● Recall how to do matrix multiplication● Dot product● Cross product is the determinant of a matrix

z

y

x

v

v

v

MMM

MMM

MMM

vM

333231

232221

131211

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Matrices

● Matrix multiplication is NOT commutative● Matrix multiplication is associative● Linear transformations can be represented as matrix

multiplications● When we talk about transformations, we will see that

all linear transformations can be represented by matrices

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Matrix Transformations

● A sequence or composition of linear transformations corresponds to the product of the corresponding matrices■ Note: the matrices to the right affect vector first■ Note: order of matrices matters!

● The identity matrix I has no effect in multiplication● Some (not all) matrices have an inverse:

vvMM

M

1

0)det(