CS 551 / 645: Introductory Computer Graphics

Mathematical Foundations

From last class…

U.S. film projectors play film at 24 fps– Projectors have a shutter to block light during

frame advance– To reduce flicker, shutter opens twice for each

frame – resulting in 48 fps flashing– 48 fps is perceptually acceptable

European film projectors play film at 25 fps– American films are played ‘as is’ in Europe,

resulting in everything moving 4% faster– Faster movements and increased audio pitch are

considered perceptually acceptable

From last class…

Film to DVD transfer– Problem: 24 film fps must be converted to television

interlaced 29.97 (30) fps

Use 3:2 Pulldown– First frame of movie is broken into first three frames– Next frame of movie is broken into next two frames– Next frame of movie is broken into next three frames…

From last class…

Additional Displays

Display Walls (Princeton)

Additional Displays

Stereo

Mathematical Foundations

Hearn and Baker (A1 – A4) appendix gives good review

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

Points, vectors, and coordinates

– Dot and cross products– Linear transforms and matrices

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

Trigonometry

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

– sin () = x– cos () = y– tan () = x/y– Etc…

(x, y)

Slope-intercept Line Equation

Slope = m = rise / run

Slope = (y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)Solve for y:

y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1

or: y = mx + b

x

y

P2 = (x2, y2)

P1 = (x1, y1)

P = (x, y)

Parametric Line Equation

Given points P1 = (x1, y1) and P2 = (x2, y2)x = x1 + t(x2 - x1)y = y1 + t(y2 - y1)

When:– t=0, we get (x1, y1)– t=1, we get (x2, y2)– (0<t<1), we get points

on the segment between(x1, y1) and (x2, y2)

x

y

P2 = (x2, y2)

P1 = (x1, y1)

Other helpful formulas

Length = sqrt (x2 - x1)2 + (y2 - y1)2

Midpoint, p2, between p1 and p3– p2 = ((x1 + x3) / 2, (y1 + y3) / 2))

Two lines are perpendicular if:– M1 = -1/M2– cosine of the angle between them is 0

Coordinate Systems

Z

X

YY

X

Z

Right-handedcoordinatesystem

Left-handedcoordinatesystem

Grasp z-axis with handRoll fingers from positive x-axis towards positive y-axisThumb points in direction of z-axis

Points

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

– Note that the addition of two points is not defined

P

Q

v

Vectors

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

Vector Spaces

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

Operations:– Addition– Subtraction– Dot Product– Cross Product– Norm

Vector Addition/Subtraction

– operation u + v, with: Identity 0 v + 0 = v Inverse - v + (-v) = 0

– Vectors are “arrows” rooted at the origin– Addition uses the “parallelogram rule”:

u+vy

xu

v

u-v

y

x

uv

-v

Scalar Multiplication

– Scalar multiplication: Distributive rule:(u + v) = (u) + (v)

( + )u = u + u

– Scalar multiplication “streches” a vector, changing its length (magnitude) but not its direction

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 (Euclidean Norm) of a vector: length(v) = ||v|| = sqrt(v • v)

– Normalizing a vector, making it unit-length: v = v / ||v||– Computing the angle between two vectors:

u • v = |u| |v| cos(θ)

– Checking two vectors for orthogonality u • v = 0.0

u θ

v

u

Dot Product

Projecting one vector onto another– If v is a unit vector and we have another vector, w– We can project w perpendicularly onto v

– And the result, u, has length w • v

w

v

wv

wv

wvw

wu

)cos(

Dot Product

Is commutative– u • v = v • u

Is distributive with respect to addition– u • (v + w) = u • v + u • w

Cross Product

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

The cross product of two vectors is orthogonal to both

Right-hand rule dictates direction of cross product

1221

1221

1221

21 )(

y x y x

z x z x

z y z y

vv

Cross Product Right Hand Rule

See: http://www.phy.syr.edu/courses/video/RightHandRule/index2.html

Orient your right hand such that your palm is at the beginning of A and your fingers point in the direction of A

Twist your hand about the A-axis such that B extends perpendicularly from your palm

As you curl your fingers to make a fist, your thumb will point in the direction of the cross product

Cross Product Right Hand Rule

See: http://www.phy.syr.edu/courses/video/RightHandRule/index2.html

Orient your right hand such that your palm is at the beginning of A and your fingers point in the direction of A

Twist your hand about the A-axis such that B extends perpendicularly from your palm

As you curl your fingers to make a fist, your thumb will point in the direction of the cross product

Cross Product Right Hand Rule

See: http://www.phy.syr.edu/courses/video/RightHandRule/index2.html

Orient your right hand such that your palm is at the beginning of A and your fingers point in the direction of A

Twist your hand about the A-axis such that B extends perpendicularly from your palm

As you curl your fingers to make a fist, your thumb will point in the direction of the cross product

Cross Product Right Hand Rule

See: http://www.phy.syr.edu/courses/video/RightHandRule/index2.html

Orient your right hand such that your palm is at the beginning of A and your fingers point in the direction of A

Twist your hand about the A-axis such that B extends perpendicularly from your palm

As you curl your fingers to make a fist, your thumb will point in the direction of the cross product

Cross Product Right Hand Rule

See: http://www.phy.syr.edu/courses/video/RightHandRule/index2.html

Orient your right hand such that your palm is at the beginning of A and your fingers point in the direction of A

Twist your hand about the A-axis such that B extends perpendicularly from your palm

As you curl your fingers to make a fist, your thumb will point in the direction of the cross product

Triangle Arithmetic

Consider a triangle, (a, b, c)– a,b,c = (x,y,z) tuples

Surface area = sa = ½ * ||(b –a) X (c-a)||

Unit normal = (1/2sa) * (b-a) X (c-a)

a c

b

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

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– If basis vectors are orthogonal and unit length:

Vectors comprise orthonormal basis

– Best-known example: Cartesian coordinates– Note that a given vector v will have different

coordinates for different bases

Conclusion

Read Chapters 1 – 3 of OpenGL Programming Guide

Assignment 1 out today, due 8 days from now, Wednesday the 12th