Geometry review, part I

Transcription

1 Geometr reie, part I Geometr reie I Vectors and points points and ectors Geometric s. coordinate-based (algebraic) approach operations on ectors and points Lines implicit and parametric equations intersections, parallel lines Planes implicit and parametric equations intersections ith lines

2 Geometr s. coordinates Geometric ie: a ector is a directed line segment, ith position ignored. different line segments (but ith the same length and direction) define the same ector A ector can be thought of as a translation. Algebraic ie: a ector is a pair of numbers Vectors and points Vector = directed segment ith position ignored + - c addition negation multiplication b number Operations on points and ectors: point - point = ector point + ector = point

3 Dot product Dot product: used to compute projections, angles and lengths. Notation: ( ) = dot product of ectors and. ( ) = cosα, α = length of Properties: if and are perpendicular, ( ) = 0 ( ) = angle beteen and : cosα= ( )/ length of projection of on : ( )/ Coordinate sstems For computations, ectors can be described as pairs (D), triples (3D), of numbers. Coordinate sstem (D) = point (origin) + basis ectors. Orthogonal coordinate sstem: basis ectors perpendicular. Orthonormal coordinate sstem: basis ectors perpendicular and of unit length. Representation of a ector in a coordinate sstem: numbers equal to the lengths (signed) of projections on basis ectors.

4 Operations in coordinates = ( e )e + ( e )e e orks onl for orthonormal coordinates! e = e + e = [, ] Operations in coordinate form: + = [, ]+ [, ]= [ +, + ] - = [ -, - ] α = = [α, α ] Dot product in coordinates α α α cosα = cos( α α = = = (cos ( + α cosα / + + sinα / ) sinα ) ) Linear properties become obious: ( (+) u) = ( u) + ( u) (a ) = a( )

5 3D ectors Same as D (directed line segments ith position ignored), but e hae different properties. In D, the ector perpendicular to a gien ector is unique (up to a scale). In 3D, it is not. To 3D ectors in 3D can be multiplied to get a ector (ector or cross product). Dot product orks the same a, but the coordinate epression is ( ) = + + Vector (cross) product has length sinα = area of the parallelogram ith to sides gien b and, and is perpendicular to the plane of and. Direction (up or don) is ( + ) u = u + u determined b ( c) = c( ) the right-hand rule. = - unlike a product of numbers or dot product, ector product is not commutatie!

6 Vector product Coordinate epressions is perpendicular to, and : ( u ) = 0 (u ) = 0 the length of u is sinα : (u u) = sin α = ( 1 cos α) = ( (,)) Sole three equations for u, u, u Vector product Phsical interpretation: torque ais of rotation torque = displacement r r F force F

7 Vector product Coordinate epression: =,det det, det e e e det Notice that if = =0, that is, ectors are D, the cross product has onl one nonero component () and its length is the determinant det Vector product More properties c(a b) b(a c) c)) a (b ( = (b c)(a d) (a c)(b d) d)) b) (c a (( =

8 Line equations Intersecting to lines: 1 1 take one in implicit form: ((q p ) n ) = 0 the other in parametric: q = p + t i i If q = p + t is the intersection point, it satisfies both equations. Plug parametric into implicit, sole for t i : If i 1 ((p + t p ) n ) = i (p p n ) ( n ) 0, then t = 1 ( n ) Otherise, the lines are parallel or coincide. 001, Denis Zorin Plane equations implicit equation: (q-p,n)=0, eactl like line in D! n q p parametric equation: parameters t 1,t q(t 1,t ) = 1 t 1 + t, here 1 and are to ectors in the plane. = n 001, Denis Zorin 1

9 Intersecting a line and a plane Same old trick: use the parametric equation for the line, implicit for the plane. p 1 n qi p (p 1 i + t p n) = 0 t i 1 (p p n) = ( n) Do not forget to check for ero in the denominator! 001, Denis Zorin Transformations Eamples of transformations: translation rotation scaling shear 001, Denis Zorin 3