3D SurfacesAdrien TreuilleCarnegie Mellon Universitysource: http://iparla.labri.fr/publications/2007/BS07b/sketch_teaser.jpgOutline•Homework 1•Height Fields-Normals•Implicit and Explicit Surfaces-Numerical vs. Analytic-Implicit vs. Explicit-Conversions•Subdivision Surfaces•Mesh Editingsource: http://www.cs.umd.edu/class/spring2005/cmsc828v/thumbnails/thMPU.gifsource: http://graphics.cs.lth.se/theses/projects/projgrid/demo11.jpgsource: http://iparla.labri.fr/publications/2007/BS07b/sketch_teaser.jpgOutline•Homework 1•Height Fields-Normals•Implicit and Explicit Surfaces-Numerical vs. Analytic-Implicit vs. Explicit-Conversions•Subdivision Surfaces•Mesh Editingsource: http://www.cs.umd.edu/class/spring2005/cmsc828v/thumbnails/thMPU.gifsource: http://graphics.cs.lth.se/theses/projects/projgrid/demo11.jpgsource: http://iparla.labri.fr/publications/2007/BS07b/sketch_teaser.jpgHomework 1Homework 115-462: Computer GraphicsFebruary 3, 20091. For what values of α is a orthogonal to b − αa? What about the specialcase where ||a|| = 1? The case where ||a|| = 0?2. Are these a pure rotation matrices? Why or why not?1√21 0 110−10 −√2013√223−44042 −3 −43. Please sketch the 2D parametric curve for the function f : [0, 1) → R2wheref =)0, 3t*Tif 0 ≤ t<13)13sin+3π(t −13),,34+14cos+3π(t −13),*Tif13≤ t<23)t −23,12−32(t −23)*Tif23≤ t<1Also please label the 2D coordinates of the endpoints. What is the normalat)0,13*? How about at t =12?(Hint: You will R ecognize this shape.)4. Please sketch the 2D implicit curve for the function f : R2→ R where:f(x, y)=|x| + |y| − 1Please label 2D coordinates as appropriate. What is the normal at)12,12,*?Does this function satisfy the inside/outside convention for implicit sur-faces? Why?5. Given parallel lines f1(t) = [x1,y1,z1]T+t [u, v, w]Tand f2(t) = [x2,y2,z2]T+t [u, v, w]T, what is their common 2D vanishing point under the perspec-tive projection p(x, y, z)=)xz,yz*T?6. Given a continuous parametric function f : R2→ R3specifying a 2Dsurface in 3D space, define a continuous implicit function g : R3→ Rcorresponding to the same surface. Notes:• You do not need to prove the continuity of your function. (But youcan for extra credit!)• You’ll like ly want to use the infimum function.1• You can ignore the inside/outside convention: g can be everywherenonnegative.1http://en.wikipedia.org/wiki/Infimum1Homework 1Homework 115-462: Computer GraphicsFebruary 3, 20091. For what values of α is a orthogonal to b − αa? What about the specialcase where ||a|| = 1? The cas e where ||a|| = 0?2. Are thes e a pure rotation matrices? Why or why not?1√21 0 110−10 −√2013√223−44042 −3 −43. Please sketch the 2D parametric curve for the function f : [0, 1) → R2wheref =)0, 3t*Tif 0 ≤ t<13)13sin+3π(t −13),,34+14cos+3π(t −13),*Tif13≤ t<23)t −23,12−32(t −23)*Tif23≤ t<1Also please label the 2D coordinates of the endpoints. What is the normalat)0,13*? How about at t =12?(Hint: You will R ecognize this shape.)4. Please sketch the 2D implicit curve for the function f : R2→ R where:f(x, y)=|x| + |y| − 1Please label 2D coordinates as appropriate. What is the normal at)12,12,*?Does this function satisfy the inside/outside convention for implicit sur-faces? Why?5. Given parallel lines f1(t) = [x1,y1,z1]T+t [u, v, w]Tand f2(t) = [x2,y2,z2]T+t [u, v, w]T, what is their common 2D vanishing point under the perspec-tive projection p(x, y, z)=)xz,yz*T?6. Given a continuous parametric function f : R2→ R3specifying a 2Dsurface in 3D space, define a continuous implicit function g : R3→ Rcorresponding to the same surface. Notes:• You do not need to prove the continuity of your function. (But youcan for extra credit!)• You’ll like ly want to use the infimum function.1• You can ignore the inside/outside convention: g can be everywherenonnegative.1http://en.wikipedia.org/wiki/Infimum1Homework 1Homework 115-462: Computer GraphicsFebruary 3, 20091. For what values of α is a orthogonal to b − αa? What about the specialcase where ||a|| = 1? The cas e where ||a|| = 0?2. Are thes e a pure rotation matrices? Why or why not?1√21 0 110−10 −√2013√223−44042 −3 −43. Please sketch the 2D parametric curve for the function f : [0, 1) → R2wheref =)0, 3t*Tif 0 ≤ t<13)13sin+3π(t −13),,34+14cos+3π(t −13),*Tif13≤ t<23)t −23,12−32(t −23)*Tif23≤ t<1Also please label the 2D coordinates of the endpoints. What is the normalat)0,13*? How about at t =12?(Hint: You will R ecognize this shape.)4. Please sketch the 2D implicit curve for the function f : R2→ R where:f(x, y)=|x| + |y| − 1Please label 2D coordinates as appropriate. What is the normal at)12,12,*?Does this function satisfy the inside/outside convention for implicit sur-faces? Why?5. Given parallel lines f1(t) = [x1,y1,z1]T+t [u, v, w]Tand f2(t) = [x2,y2,z2]T+t [u, v, w]T, what is their common 2D vanishing point under the perspec-tive projection p(x, y, z)=)xz,yz*T?6. Given a continuous parametric function f : R2→ R3specifying a 2Dsurface in 3D space, define a continuous implicit function g : R3→ Rcorresponding to the same surface. Notes:• You do not need to prove the continuity of your function. (But youcan for extra credit!)• You’ll like ly want to use the infimum function.1• You can ignore the inside/outside convention: g can be everywherenonnegative.1http://en.wikipedia.org/wiki/Infimum1Homework 1Homework 115-462: Computer GraphicsFebruary 3, 20091. For what values of α is a orthogonal to b − αa? What about the specialcase where ||a|| = 1? The cas e where ||a|| = 0?2. Are thes e a pure rotation matrices? Why or why not?1√21 0 110−10 −√2013√223−44042 −3 −43. Please sketch the 2D parametric curve for the function f : [0, 1) → R2wheref =)0, 3t*Tif 0 ≤ t<13)13sin+3π(t −13),,34+14cos+3π(t −13),*Tif13≤ t<23)t −23,12−32(t −23)*Tif23≤ t<1Also please label the 2D coordinates of the endpoints. What is the normalat)0,13*? How about at t =12?(Hint: You will R ecognize this shape.)4. Please sketch the 2D implicit curve for the function f : R2→ R where:f(x, y)=|x| + |y| − 1Please label 2D coordinates as appropriate. What is the normal at)12,12,*?Does this function satisfy the inside/outside convention for implicit
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