Lecture 12 - Spline surfaces (slides)

# A b c d e f there are six unknown parameter vectors

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Unformatted text preview: e⌫ + f P(µ, ⌫ ) = aµ2 + d⌫ 2 + 2bµ⌫ + 2cµ + 2e⌫ + f There are six unknown parameter vectors •  There are six unknown parameter vectors !{a, b, c, d, e, f } There are six unknown parameter vectors {a, b, c, d, e, f } Graphics Lecture 12: Slide 6! 6 / 35 Associating points and parameters Associating points and parameters " We can solve for the six vector unknowns by substituting in six points at can solve for of µ six vector unknowns by substituting •  We known values the and ⌫ . in six points at known values of µ and ν. ! We   might have an arrangement such as: such as: ! • We might have an arrangement P0 P1 P2 P3 P4 P5 µ 0 0 1 1 1/2 1/2 ⌫ 0 1 0 1 0 1 Graphics Lecture 12: Slide 7! 7 / 35 Surface parameter equations Surface parameter equations" Substituting thesethese values of ⌫ into thento theequation gives •  Substituting values of µ and µ and ν i patch patch us these six equations these six equations ! equation gives us P0 = f P1 = d + 2 e + f P2 = a + 2 c + f P3 = a + 2 b + 2c + d + 2 e + f P4 = a/4 + c + f P5 = a/4 + b + c + d + 2e + f The The are known and we can solve for the unknowns {a, . . . , f } •  P’s P’s are known and we can solve for the unknowns using a, . . . , f} methods: { standard using standard methods! Graphics Lecture 12: Slide 8! 8 / 35 Getting the edges from the surface equation Getting the edges from the surface equation P(µ, ⌫ ) = aµ2 + d⌫ 2 + 2bµ⌫ + 2cµ + 2e⌫ + f (µ, ⌫ ) = aµ2 + d⌫ 2 + 2bµ⌫ + 2cµ + 2e⌫ + f P Getting the edges from the surface equation µ and ⌫ are in the range [0, 1]. ⌫ are in range [0 1] µ and and ν arethethe range ,[0, .1]. in Thusµthe contours that bound the contours that bound ThusThus the contours that bound the the patch can be found by found by the patch can be found by patch can substituting 0 or 1 for one of µ µ for one of µ or substituting substituting 0 or01or 1 for one of oror ⌫ in the the patch equation. ν in patch equation. ⌫ in the patch equation. ! ! P(0, ⌫ ) P(0, ⌫ ) P(1, ⌫ ) P(1, ⌫ ) P(µ, 0) P(µ, 0) P(µ, 1) P(µ, 1) = = = = = = = = Graphics Lecture 12: Slide 9! " 2 d⌫ 2 + 2e⌫ + f d⌫ + 2e⌫ + f 2 a + 2(b + e)⌫ + 2c +...
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