11 geometry

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Geris Game Pixar.The Lord of the Rings New Line Productions, Inc.Finding Nemo Disney/Pixar.

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This Sections Objectives

Define the basic elements of geometry

Scalars

Vectors

Develop mathematical operations among them in acoordinate-freemanner

e ne as c grap cs pr m ves Line segments

Polygons

CSCI 3161- Com uter Animation

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Basic Elements

Geometry is the study of the relationshipsamong o ects n an n- mens ona space we are interested in objects that exist in (at most) 3D

We want a minimum set of primitives from which

we can build more sophisticated objects

We will need three basic elements Scalars

Vectors

Points


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Coordinate-Free Geometry

When we learned simple geometry, most of us

started with a Cartesian approach Points were at locations in space P=(x,y,z)

involving these coordinates

Most geometric results are independent of the

coordinate system

corresponding sides and the angle between them are

identical


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Scalars

Scalars are members of sets that can be combinedy a on an mu p ca on o ey ng some

fundamental axioms

a ociati it : x * * z = x * * z

commutativity : x + y = y + x

distributivity : x * (y + z) = (x * y) + (x * z) : - =

multiplicative inverse : x * (1/x) = 1

Scalars alone have no geometric properties


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Vectors

A vector is an object with two attributes

Direction Magnitude

Examples include v Force

Velocity

rec e ne segmen s Most important example for graphics


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Vector Operations

Every vector has an inverse

Same magnitude but points in opposite direction

Every vector can be multiplied by a scalar

There is a zero vector

Zero magnitude, undefined orientation

The sum of any two vectors is a vector

v -v 0.7v v w


u

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Linear Vector Spaces

Mathematical system for manipulating vectors

Operations Scalar-vector multiplication : u = v - w u v

Expressions such as v=u+2w-3rmake sense in a vector space

Magnitude of a vector

Calculate length with Pythagorean theorem

Normalizing a vector

Form a unit vector


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Linear Combinations & Independence

A linear combination of n vectors u1, u2,,un is the

u = 1u1 + 2u2 ++ nun

If the only set of scalars that produces

1 1 2 2 n n

is

1 = 2 == n =


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Basis Vectors & Representation

If a vector space has dimension n, any set of n linearly

If u1, u2,,un is a basis for vector space, V, any vector vcan

vectors:

v = u + u ++ u

The scalars i give the representation ofv with respect to the basisu , u ,,u :

[ ]

=Tn

M

K2

1

21


n

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Inner (Dot) Product

The inner (dot) product between vectors uand vin an-

n

ii vu )*(

if , then u and v are orthogonal (rightangles in 2D)

=i 1

0= vu

length of vector uis: uuu =

can use the dot product to define the angle ( )between two vectors:


cosvuvu =

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Projections of Vectors

The dot product can also be used to project vector

A onto vector B, yielding a new vector C

where

BB

is the unit vector of BBu


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Projecting Vector A on Vector B

#include #include

.

// a point data type

t edef GLfloat oint2[2];

// the vectors

point2 vectorA = {0.0, 0.0};

point2 vectorB = {0.0, 0.0};point2 vectorC = {0.0, 0.0};


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void initializeGL(void)

{

// enter GL_PROJECTION mode

glMatrixMode(GL_PROJECTION);

c ear e pro ec on ma r x

glLoadIdentity();

gluOrtho2D(-1.0, 1.0, -1.0, 1.0);

setu the vectors

setupVectors();

}


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void myDisplay(void)

{

glClear(GL_COLOR_BUFFER_BIT);

glBegin(GL_LINES); // draw the lines

// draw vector A red

glColor3f( , 0, 0);

glVertex2f(0, 0);

glVertex2fv(vectorA);

draw vector B green

// draw vector C blue

glEnd();


g F us ;

}

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Example projectVector.c


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Vectors Lack Position

These vectors are identical: Same length and magnitude

Vectors s aces are insufficient for constructin

geometric objects We need points too


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Points

Location in space

perat ons a owe etween po nts an

vectors

o n -po n su rac on y e s a vec or

Point-vector addition yields a point

v= P-Qv

P = v+Q


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Affine Spaces

Point + a vector space

O erations

Vector-vector addition

Scalar-vector multiplication Scalar-scalar operations

Point-vector addition Point-Point subtraction


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Lines

Consider all points of the form: P()=P0 + v

Set of all points that pass through P0 in thedirection of the vector v or -v

P()

v

P0P(-)


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Rays

For 0, P() is the rayleaving P in

the direction v :

P()

v

P0


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Line Segments

For 0 1 we get all the points on the

line segmentjoining P0 and P(1.0) :

.

v

P0


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Parametric Lines - Interpolation

Nice since it extends to curves and

surfaces Forms

Explicit:

y = mx + h Parametric:

x() = x0 + (1-)x1P1= (x1, y1)

0 - 1P0= (x0, y0)


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Convexity

An object is convexiff:

for any two points in the object all points on the line

segment between these points are also in the object

P

P

Q

-


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Triangles

3 sided polygon

Can be defined by:

3 points

2 points and 2 angles


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Triangles Bi-linear Interpolation

interpolation of

P and Q


,

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Triangles Bi-linear Interpolation

interpolation of

,


,

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Polygons

Definition: a closed plane figure bounded by

straight sides

Can always be broken up into triangles


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Polygons

Special names of common polygons:

triangle A polygon with three sides

quadrilateral A polygon with four sides

pentagon A polygon with five sideshexagon A polygon with six sides

heptagon A polygon with seven sides

oc agon po ygon w e g s esn-gon A polygon with n sides


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Planes in 3D

A plane can be determined by:

a point (R) and two vectors (mand n)

m

Rn


, = + m + n

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Planes in 3D

Or equivalently by:

three points (m= P-R and n= Q-R)

P

R Q


P(, ) = R + - + -

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Normals

Every plane has a vector normal n(perpendicular) to it

we can use the cross product to find n

n

u

v


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The Cross Product

To compute the cross product between 3D vectors

uand v:

xyyxzxxzyzzy vuvuvuvuvuvuvu = ,,

n

u

v


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Questions


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