SER332 Lecture 13

jgs SER332 Introduction to Graphics and Game Development Lecture 13:
Linear Algebra Javier Gonzalez-Sanchez [email protected] PERALTA 230U Office Hours: By appointment

Javier Gonzalez-Sanchez | SER332 | Spring 2018 | 1 jgs
Announcement § Project 2 is Posted Due by March 21, 11:59PM

Goal

Goal 3D Maze ( IJKS 1982)

jgs Points, Vector, Angles Linear Algebra

jgs Javier Gonzalez-Sanchez | SER332 | Spring 2018 | 6
Points Specify locations in space (or in the plane) Points and Vectors Vector have a magnitude and direction (4,2) (1,3) v ⍬

Vectors § Vector v = (v1 , v2 ) § Magnitude or Length ||v|| = sqrt( v1 2 + v2 2) § Orientation (angle) ⍬ = tan -1 ( v2 / v1 ) § Unit vector ||v|| = 1 § Zero vector v = (0,0)

u - v = (u1 ,u2 ) - (v1 ,v2 ) = (u1 -v1 , u2 -v2 ) u + v = (u1 ,u2 ) + (v1 ,v2 ) = (u1 +v1 , u2 +v2 ) Addition and Subtraction v u u+v v u v-u

a * v = a * (v1 ,v2 ) = (a*v1 , a*v2 ) Multiplication with a Scalar v av

Test yourself § Point + Vector = ? § Point – Point = ? § Vector + Vector = ? § Point + Point = ?

Test yourself § Point + Vector = Point § Point – Point = Vector § Vector + Vector = Vector § Point + Point = ? v u u-v v u v-u

Angles

Uses // mouse motion callback function void motion(int x, int y) { if (mouse_button == GLUT_LEFT_BUTTON) { y_angle += (float(x - mouse_x) / width) *360.0; x_angle += (float(y - mouse_y) / height)*360.0; } if (mouse_button == GLUT_RIGHT_BUTTON) { scale += (y - mouse_y) / 100.0; if (scale < 0.1) scale = 0.1; if (scale > 7) scale = 7; } mouse_x = x; mouse_y = y; glutPostRedisplay(); }

jgs Dot product

§ The dot product or inner product measures to what degree two vectors are aligned u · v = (u1 ,u2 ) · (v1 ,v2 ) = u1 * v1 + u2 * v2 v * u = (5*7) + (5*3) = 50 v * w = (5*5) + (5*0) = 25 u * w = (7*5) + (3*0) = 35 u * u = (7*7) + (3*3) = 58 t * w = (0*5) + (5*0) = 0 Dot Product q v=[5,5] u=[7,3] w=[5,0] t=[0,5]

§ Two vectors v and w are perpendicular or normal iff u · v = 0 § Geometric Interpretation: § cos(a) = l / ||w|| = u · v / ||v|| * ||w|| Dot Product w v a l

v * u = (5*7) + (5*3) = 50 v * w = (5*5) + (5*0) = 25 u * w = (7*5) + (3*0) = 35 u * u = (7*7) + (3*3) = 58 t * w = (0*5) + (5*0) = 0 |t| = 5.0 |v| = 7.07 |u| = 7.61 |w| = 5.0 § cos(a) = u · v / ||v|| * ||w|| q = cos-1 (25 / 7.07 * 5.0) = 45º Dot Product q v=[5,5] u=[7,3] w=[5,0] t=[0,5]

Dot Product ||v|| = sqrt (v · v) = sqrt (v1 * v1 + v2 * v2 )

Dot Product | Uses § Calculate the angle between 2 vectors

jgs Cross product

• Notation: u = v x w • The cross product is a vector • ||u|| proportional to the sine of the angle between v and w ||u|| = ||v||*||w||*sin(a) • u is perpendicular to v and w • The direction of u follows the right hand rule Cross Product w v a u

Cross Product u = v x w = ! " ! # ! $ x % " % # % $ = & ' ( ! " ! # ! $ % " % # % $ = ! # ∗ % $ − % # ∗ ! $ ! $ ∗ % " − % $ ∗ ! " ! " ∗ % # − % " ∗ ! #

Cross Product | Uses § Calculate the area of a triangle § Calculate distance from a line to a dot § Calculate normals

Cross Product | Triangle Area § The cross product is related to the area of a triangle § (parallelogram)area = ||u|| = ||v x w|| = ||v||*||w||*sin(a) § (triangle)area = ||u|| / 2 = ||v x w|| / 2 = ... Q R P w v

Test Yourself Given a Triangle with P=(0,0,0), Q=(0,3,0) and R=(3,3,0) What is the area? Q R P

Test Yourself § area (triangle) = (0,3,0) x (3,3,0) = ||(0,0,-9)|| / 2 = 9/2 = 4.5 § area (parallelogram) = 9 § area (parallelogram) = (3)(4.24) sin (45) = (3)(4.24...)(0.70...) = 9 Q R P

Cross Product | Distance point to line // compute the distance from AB to C double lineToPointDistance2D (double[] pA, double[] pB, double[] pC) { double dist = cross(pA-pB, pA-pC) / (pA - pB); return Math.abs(dist); } pA pC pB

Cross Product | Distance point to line

Cross Product | Normal calculation (lighting)

jgs Matrices Linear Algebra

Matrix § A matrix is a set of elements, organized into rows and columns § addition: ! " # $ + & ' ( ℎ = ! + & " + ' # + ( $ + ℎ § subtraction: ! " # $ − & ' ( ℎ = ! − & " − ' # − ( $ − ℎ § multiplication: ! " # $ − & ' ( ℎ = !& + "( !' + "ℎ #& + $( #' + $ℎ

Matrix Multiplication § Cn x p = A n x m B m x p § Examples 2 5 3 1 ∗ 6 2 1 5 = 17 29 19 11 6 2 1 5 ∗ 2 5 3 1 = 18 32 17 10

Usage: transformations

Homework Review Linear Algebra

jgs SER332 Introduction to Graphics Javier Gonzalez-Sanchez [email protected] Spring 2018
Disclaimer. These slides can only be used as study material for the class SER332 at ASU. They cannot be distributed or used for another purpose.

SER332 Lecture 13

SER332 Lecture 13

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