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Geometry review, part I 2005, Denis Zorin Geometry review I Vectors and points Q points and vectors Q Geometric vs. coordinate-based (algebraic) approach Q operations on vectors and points Lines Q implicit and parametric equations Q intersections, parallel lines Planes Q implicit and parametric equations Q intersections with lines 2005, Denis Zorin 13 Geometry vs. coordinates Geometric view: a vector is a directed line segment, with position ignored. different line segments (but with the same length and direction) define the same vector A vector can be thought of as a translation. Algebraic view: a vector is a pair of numbers 2005, Denis Zorin Vectors and points Vector = directed segment with position ignored w -w v v+w addition w negation w cw multiplication by number Operations on points and vectors: point - point = vector point + vector = point 2005, Denis Zorin 14 Dot product Dot product: used to compute projections, angles and lengths. Notation: (w·v) = dot product of vectors w and v. w α (w·v) = |w| |v| cosα, |v| = length of v v Properties: if w and v are perpendicular, (w·v) = 0 (w·w) = |w|2 angle between w and v: cosα = (w·v)/|w||v| length of projection of w on v: (w·v)/|v| 2005, Denis Zorin Coordinate systems For computations, vectors can be described as pairs (2D), triples (3D), … of numbers. Coordinate system (2D) = point (origin) + 2 basis vectors. Orthogonal coordinate system: basis vectors perpendicular. Orthonormal coordinate system: basis vectors perpendicular and of unit length. Representation of a vector in a coordinate system: 2 numbers equal to the lengths (signed) of projections on basis vectors. 2005, Denis Zorin 15 Operations in coordinates v = (v·ex)ex + (v·ey)ey ey vy vx works only for orthonormal coordinates! v v = vx ex + vy ey = [ vx, vy ] ex Operations in coordinate form: v + w = [ vx, vy ] + [ wx, wy ] = [vx + wx, vy + wy] -w = [ -wx, -wy ] α w = = [α wx, α wy ] 2005, Denis Zorin Dot product in coordinates | v || w | cos α =| v || w | cos(α v − α w ) =| v || w | (cos α v cos α w + sin α v sin α w ) vy wy v =| v || w | (v x w x / | v || w | + v y w y / | v || w |) α αw αv vx = vx w x + vyw y w wx Linear properties become obvious: ( (v+w)·u) = (v·u) + (w·u) (av·w) = a(v·w) 2005, Denis Zorin 16 3D vectors Same as 2D (directed line segments with position ignored), but we have different properties. In 2D, the vector perpendicular to a given vector is unique (up to a scale). In 3D, it is not. Two 3D vectors in 3D can be multiplied to get a vector (vector or cross product). Dot product works the same way, but the coordinate expression is (v·w) = vxwx+ vywy + vzwz 2005, Denis Zorin Vector (cross) product v × w has length v w sin α w v = area of the parallelogram with two sides given by v and w, and is perpendicular to the plane of v and w. ( v + w ) × u = v × u + w × u Direction (up or down) is (cv) × w = c( v × w ) v×w = - w ×v 2005, Denis Zorin determined by the right-hand rule. unlike a product of numbers or dot product, vector product is not commutative! 17 Vector product Coordinate expressions v × w is perpendicular to v, and w: (u·v) = 0 (u·w ) = 0 the length of u is v w sin α : 2 2 2 2 (u·u) = v w sin2 α = v w (1 − cos2 α) = v w ( v w − (v, w )) Solve three equations for ux, uy, uz 2005, Denis Zorin Vector product Physical interpretation: torque torque = r×F axis of rotation force F displacement r 2005, Denis Zorin 18 Vector product Coordinate expression: ex det v x w x ey vy wy ez vy v z = det w y w z vz v ,− det x wz w x vz vx , det wz w x vy w y Notice that if vz=wz=0, that is, vectors are 2D, the cross product has only one nonzero component (z) and its length is the determinant vx vy det w x w y 2005, Denis Zorin Vector product More properties (a·(b × c)) = b(a·c) − c(a·b) ((a × b)·(c × d)) = (a·c)(b·d) − (b·c)(a·d) 2005, Denis Zorin 19 Line equations Intersecting two lines: take one in implicit form:((q − p1)· n1) = 0 the other in parametric: q = p2 + v2 t If qi = p2 + v2 t i is the intersection point, it satisfies both equations. Plug parametric into implicit, solve for ti : ((p2 + v2 ti − p1)·n2 ) = 0 If (v2 ·n1) ≠ 0 , then t i = − (p2 − p1·n1) (v 2 · n1) Otherwise, the lines are parallel or coincide. © 2001, Denis Zorin Plane equations implicit equation : (q-p,n)=0, exactly like line in 2D! n q p parametric equation: 2 parameters t1,t2 q(t1,t2) = v1 t1 + v2 t2, where v1 and v2 are two vectors in the plane. v1 × v 2 = n © 2001, Denis Zorin 2 Intersecting a line and a plane Same old trick: use the parametric equation for the line, implicit for the plane. p1 v n qi p2 (p1 + vti − p2 · n) = 0 (p1 − p2 · n) Do not forget to check t =− for zero in the denominator! (v · n) i © 2001, Denis Zorin Transformations Examples of transformations: translation rotation scaling shear © 2001, Denis Zorin 3