3D Geometry for Computer Graphics Class 1

General Office hour: Sunday 11:00 – 12:00 in Schreiber 002 (contact in advance) Webpage with the slides:

The plan today Basic linear algebra and Analytical geometry

Why??

We represent objects using mainly linear primitives:  points  lines, segments  planes, polygons Need to know how to compute distances, transformations, projections…

Basic definitions Points specify location in space (or in the plane). Vectors have magnitude and direction (like velocity). Points  Vectors

Point + vector = point

vector + vector = vector Parallelogram rule

point - point = vector A B B – A A B A – B

point + point: not defined!!

Map points to vectors If we have a coordinate system with origin at point O We can define correspondence between points and vectors:

Inner (dot) product Defined for vectors:  L v w Projection of w onto v

Dot product in coordinates (2D) v w xvxv yvyv xwxw ywyw x y O

Perpendicular vectors (2D) v vv

Parametric equation of a line p0p0 v t > 0 t < 0 t = 0

Parametric equation of a ray p0p0 v t > 0 t = 0

Distance between two points B A xBxB yByB xAxA yAyA x y O

Distance between point and line Find a point q’ such that (q  q’)  v dist(q, l) = || q  q’ || p0p0 v q q’ = p 0 +tv l

Easy geometric interpretation p0p0 v q q’ l L

Distance between point and line – also works in 3D! The parametric representation of the line is coordinates-independent v and p 0 and the checked point q can be in 2D or in 3D or in any dimension…

Implicit equation of a line in 2D x y Ax+By+C > 0 Ax+By+C < 0 Ax+By+C = 0

Line-segment intersection x y Ax+By+C > 0 Ax+By+C < 0 Q 1 (x 1, y 1 ) Q 2 (x 2, y 2 )

Representation of a plane in 3D space A plane  is defined by a normal n and one point in the plane p 0. A point q belongs to the plane  = 0 The normal n is perpendicular to all vectors in the plane n p0p0 q 

Distance between point and plane Project the point onto the plane in the direction of the normal: dist(q,  ) = ||q’ – q|| n p0p0 q’  q

Distance between point and plane n p0p0 q’  q

Implicit representation of planes in 3D (x, y, z) are coordinates of a point on the plane (A, B, C) are the coordinates of a normal vector to the plane Ax+By+Cz+D > 0 Ax+By+Cz+D < 0 Ax+By+Cz+D = 0

Distance between two lines in 3D p1p1 p2p2 u v d The distance is attained between two points q 1 and q 2 so that (q 1 – q 2 )  u and (q 1 – q 2 )  v q1q1 q2q2 l1l1 l2l2

Distance between two lines in 3D p1p1 p2p2 u v d q1q1 q2q2 l1l1 l2l2

p1p1 p2p2 u v d q1q1 q2q2 l1l1 l2l2

p1p1 p2p2 u v d q1q1 q2q2 l1l1 l2l2

Barycentric coordinates (2D) Define a point’s position relatively to some fixed points. P =  A +  B +  C, where A, B, C are not on one line, and , ,   R. ( , ,  ) are called Barycentric coordinates of P with respect to A, B, C (unique!) If P is inside the triangle, then  +  +  =1, , ,  > 0 A B C P

Barycentric coordinates (2D) A B C P

Example of usage: warping

Tagret A B C We take the barycentric coordinates , ,  of P’ with respect to A’, B’, C’. Color( P ) = Color(  A +  B +  C )

See you next time!