Geometric Intuition Randy Gaul Talk Outline Vectors Points

Geometric Intuition Randy Gaul

Talk Outline • • • Vectors, Points and Basis Matrices Rotation Matrices Dot product and how it’s useful Cross product and how it’s useful Extras

Prerequisites • • • Matrix multiplication How to apply dot/cross product Vector normalization Basic understanding of sin/cos Basic idea of what a plane is

Vectors • Formal definition – Input or output from a function of vector algebra • Informal definition – Scalar components representing a direction and magnitude (length) – Vectors have no “location”

Points and Vectors • A point P and Q is related to the vector V by: – P – Q = V • This implies that points can be translated by adding vectors

Points and Vectors (2) • Lets define an implied point O – O represents the origin • Any point P can be expressed with a vector V: – P = O + V • Vectors point to points, when relative to the origin!

Euclidean Basis • Standard Euclidean Basis (called E 3) – Geometrically the x, y and z axes

Euclidean Basis (2) • Given 3 scalars i, j and k, any point in E 3 can be represented

Linear Combination • Suppose we have a vector V – Consists of 3 scalar values – Below V written in “shorthand” notation

Linear Combination (2) • V can be written as a linear combination of E 3 – This is the “longhand” notation of a vector

Basis Matrices • E 3 is a basis matrix – Many matrices can represent a basis • Any vector can be represented in any basis Note: Different i, j and k values are used on left and right http: //en. wikipedia. org/wiki/Change_of_basis

Rotation Matrices • A rotation matrix can rotate vectors – Constructed from 3 orthogonal unit vectors – Can be called an “orthonormal basis” • E 3 is a rotation matrix!

Rotation Matrices (2) • Rotation matrices consist of an x, y and z axis – Each axis is a vector X y z

Rotation Matrices (3) • Multiplying a rotation and a vector rotates the vector • This is a linear combination

Dot Product • Comes from law of cosines • Full derivation here • For two vectors u and v:

Shortest Angle Between 2 Vectors • Assume u and v are of unit length • Result in range of 0, 1 • No trig functions required

How Far in a Given Direction? • Given point P and vector V – How far along the V direction is P? x P V y

How Far in a Given Direction? (2)

Planes • Here’s the 3 D plane equation

Planes • Here’s the 3 D plane equation • WAIT A SECOND

Planes • Here’s the 3 D plane equation • WAIT A SECOND • THAT’S THE DOT PRODUCT

Planes (2) • a, b and c form a vector, called the normal • d is magnitude of the vector – Represents distance of plane from origin

Planes (3)

Planes and the Dot Product

Planes and the Dot Product (2) x plane P y

Signed Distance P to Plane

Project P onto Plane x plane P y

Rotation Matrices and Dot Product • Given matrices A and B • A * B is to dot the rows of A with columns of B • Lets assume A and B are rotation matrices http: //en. wikipedia. org/wiki/Matrix_multiplication

Rotation Matrices and Dot Product (2)

Rotation Matrices and Dot Product (3)

Rotation Matrices and Dot Product (4)

Rotation Matrices and Dot Product (5)

Point in Convex Hull • Test if point is inside of a hull – Compute plane equation of hull faces – Compute distance from plane with plane equation – If all distances are negative, point in hull – If any distance is positive, point outside hull • Works in any dimension • Can by used for basic frustum culling

Point in OBB • An OBB is a convex hull! – Hold your horses here… • Lets rotate point P into the frame of the OBB – OBB is defined with a rotation matrix, so invert it and multiply P by it – P is now in the basis of the OBB • The problem is now point in AABB

Point in Cylinder • Rotate cylinder axis to the z axis • Ignoring the z axis, cylinder is a circle on the xy plane • Test point in circle in 2 D – If miss, exit no intersection • Get points A and B. A is at top of cylinder, B at bottom • See if point’s z component is less than A’s and greater than B’s – Return intersection • No intersection

Bumper Car Damage? • Two bumper cars hit each other • One car takes damage • How much damage is dealt, and to whom?

Bumper Car Damage Answer • Take vector from one car to another, T • Damage dealt: – 1. 0 - Abs( Dot( velocity, T ) * collision. Damage )

http: //www. ra ndygaul. net/20 14/07/23/dista nce-point-toline-segment/

Visualization in 2 D P C

Cross Product • Operation between vectors • Produces a vector orthogonal to both input vectors

Cross Product Handedness http: //en. wikipedia. org/wiki/Cross_product

Cross Product Details http: //upload. wikimedia. org/wikipedia/commons/thumb/6/6 e/Cross_product. gif/2 20 px-Cross_product. gif

Plane Equation from 3 Points • • Given three points A, B and C Calculate normal with: Cross( C – A, B – A ) Normalize normal Compute offset d: dot( normal, A (or B or C) )

• Compute distance A to Plane: da • Compute distance B to Plane: db • If da * db < 0 – Intersection = A + (da / (da – db)) * (B – A) • Else – No intersection

normal P

Affine Transformations • Given matrix A, point x and vector b • An affine transformation is of the form: – Ax + b

Affine Transformations • Given 3 x 3 matrix A, point x and vector b • An affine transformation is of the form: – Ax + b • With a 4 x 4 matrix we can represent Ax + b in block formation:

Translation • Construct an affine transformation such that when multiplied with a point, translates the point by the vector b • I is the identity matrix, and means no rotation (or scaling) occurs

Translation (2) • The 1 is important, it means it is a point • If we had a zero here b wouldn’t affect x

Translation (3) • Proof that translation doesn’t affect vectors: – Translate from P to Q by T • • P – Q = T = (P + T) – (Q + T) = (P – Q) + (T – T) = P – Q

Rotation • Orthonormal basis into the top left of an affine transformation, without any translation vector:

Scaling • • • Given scaling vector S Take the matrix A Scale A’s x axis’ x component by S 0 Scale A’s y axis’ y component by S 1 Scale A’s z axis’ z component by S 2

Scaling (2)

Camera - Look. At

Camera – Look. At (2) • Won’t work when player tries to look straight up or down – Parallel vectors crossed result in the zero vector • Possible solutions: – Snap player’s view away from up/down – Use an if statement and cross with a different up vector – More solutions exist!

Barycentric Coordinates • Slightly out of scope of this presentation • See Erin Catto’s GDC 2010 lecture • Idea: – Like a linear combination, try affine combinations of points • Useful for: – Voronoi region identification, collision detection, certain graphics or shader effects