by YoonSik Shim Lines 2 D Slope intersept

by Yoon-Sik Shim Lines – 2 D § Slope intersept form y = mx + b slope : m = (x 2 -x 1) / (y 2 -y 1) § Point-Slope form y – y 1 = m(x – x 1) § Two-point form (y – y 1) / (x – x 1) = (y 2 – y 1) / (x 2 – x 1) § Intersept form (x / a) + (y / b) = 1 cgvr. korea. ac. kr Graphics Lab. / Korea Univ. 1

by Yoon-Sik Shim Lines – 2 D § Parametric equations x = au + b y = cu + d u : parametric variable x = (x 2 – x 1)u + x 1 y = (y 2 – y 1)u + y 1 for 2 D : These are Linear parametric equations § Angle ø between two lines in the plane tan ø = (m 1 -m 2) / (1+m 1 m 2) m 1 = m 2 : parallel m 1 m 2 = -1 : perpendicular cgvr. korea. ac. kr Graphics Lab. / Korea Univ. 2

by Yoon-Sik Shim Lines – 3 D § Parametric equations x = ax u + bx y = ay u + b y z = az u + bz u [0, 1] : interval (unit interval) x = (x 1 -x 0)u + x 0 y = (y 1 -y 0)u + y 0 z = (z 1 -z 0)u + z 0 u [0, 1] L = [ (x 1 -x 0)2 + (x 1 -x 0)2 ] cgvr. korea. ac. kr Length Graphics Lab. / Korea Univ. 3

by Yoon-Sik Shim Lines – 3 D § Direction cosines dx = (x 1 -x 0) / L dy = (y 1 -y 0) / L dz = (z 1 -z 0) / L P 1 dx 2 + d y 2 + d z 2 = 1 L P 0 cgvr. korea. ac. kr Graphics Lab. / Korea Univ. 4

by Yoon-Sik Shim Lines – 3 D § Computing points on a Line u = 1/n , 2/n , 3/n , ………. , (n-2)/n , (n-1)/n u = 1/n xi = (x 1 -x 0)ui + x 0 xi+1 = (x 1 -x 0)(ui+ u) + x 0 xi+1 = xi + (x 1 -x 0) u xi+1 = xi + x cgvr. korea. ac. kr Graphics Lab. / Korea Univ. 5

by Yoon-Sik Shim Lines – 3 D § Point and Line Relations ux = (x – x 0) / (x 1 – x 0) uy = (y – y 0) / (y 1 – y 0) uz = (z – z 0) / (z 1 – z 0) point : q(x, y, z) ux = uy = uz : q is on the line x 1 , y 1 p x 0 , y 0 q to implicit equation on 2 D(plane) f(x, y) = (x – x 0) / (y 1 – y 0) - (y – y 0) / (x 1 – x 0) point : p(x, y) , q(x, y) if f(p)f(q) > 0 : p and q is same side of the line f(x, y) = 0 : the point is on the line cgvr. korea. ac. kr Graphics Lab. / Korea Univ. 6

by Yoon-Sik Shim Lines – 3 D § Intersection of Lines (x 1 a – x 0 a)ua + x 0 a = (x 1 b – x 0 b)ub + x 0 b (y 1 a – y 0 a)ua + y 0 a = (y 1 b – y 0 b)ub + y 0 b (z 1 a – z 0 a)ua + z 0 a = (z 1 b – z 0 b)ub + z 0 b : intersect a b cgvr. korea. ac. kr Graphics Lab. / Korea Univ. 7

by Yoon-Sik Shim Lines – 3 D § Translating and Rotating of Lines x’ 0 = x 0 + x. T y’ 0 = y 0 + y. T x’ 1 = x 1 + x. T y’ 1 = y 1 + y. T x’ 0 = x 0 cosø - y 0 sinø y’ 0 = x 0 sinø + y 0 cosø cgvr. korea. ac. kr x’ 1 = x 1 cosø - y 1 sinø x’ 1 = x 1 sinø + y 1 cosø Graphics Lab. / Korea Univ. 8