Assume a bivariate patch p(u, v) over the unit square [0, 1]² that is given as a tensor product patch where u-sections (u fixed to some constant û; v varying across [0, 1]) are quadratic polynomials Pu:û(v) = p(û, v) while v-sections are lines pv:ô (u) = p(u, v). The boundary lines pv:o(u) and pv:1 (u) are specified by their end points p(0,0) 0.8 and p(1,0) 0.2 as well as p(0, 1) 0.3 and p(1, 1) = 0.8. The boundary quadratics pu:o(v) and pu:1 (v) interpolate p(0,0.5) = 0.1 and p(1, 0.5) = 0.9 in addition to the above given four corner-values. = = = Use Pu:û(v) = (1, v, v² ) Mq (Pu:û(0), Pu:û (0.5), Pu:û(1)) with Ma = 1 0 0 -3 4-1 2 4 2 (Pv:ô as well as pu: (u) = (1, u) M₁ (pv:v (0), P: (1)) with M₁ = = (19) 0 to formulate p(u, v) using the "geometric input" G with G = = (P(0,0%) p(0,0) p(0,0.5) p(0,1) ) = ( 0.39 0.8 0.1 0.3 0.2 0.9 0.8 p(1,0) p(1, 0.5) p(1, 1) See the figure below for (left) a selection of iso-lines of p(u, v) and (right) a 3D rendering of p(u, v) as a height surface over [0, 1]2. Black lines correspond p-levels of "/10 with n ≥ {1, 2, ………‚ 9}. 1.00 0.95 0.90 0.85 0.80 0.75 -0.15 0.70- 0.65 0.60 0.55 0.50 0.45- 0.40 0.35 15 0.30- 0.25 0.25 -0.35 Contoured patch 0.65 -0.85 0.75 0.65 0.9 0.95 0.90 0.7 0.85 0.80 0.75 0.70 0.6 0.65 0.60 0.55 0.5 p(u, v) 0.50 0.45 0.40 0.35 0.4 0.30 0.25 Patch as height over 2D 2.65° 70° 75° 80° 85°.209.85.00 0.00 0.15 0.3 0.10 0.05 0.00 1.000.95 0.90 0.850.80, 0.10 0.00 0.00 0.25 0.35 -0.55 0.10 0.6 0.55 0.5 0.20 0.45 0.45 0.15 0.5 0.6 0.55 0.5 0.5 0.45 -0.45 0.4 0.1 0.55 0.35 0.05 0.65 0.7 0.3 0.75 0.00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 -0.25- 0.95 1.00 0 น What's the "five factors formula" for p(u, v) and what's the according formula for Vp|u,v? Verify, by evaluating p(0, 1) as well as p(1, 0), that you indeed get the right values as given above “back out”, i.e., 0.3 and 0.2, respectively. Show your according calculations. = (Uinit, Vinit) T T = - Xinit Next, consider domain location Xinit (1, 0.5). What's p(xinit)? What's VP|xinit? Do one gradient descent step according to ✗next = Xinit TVPx, including a line-search for the best-possible 7. What 7-value reduces p the most? You can round this 7-value to 3 digits after the comma. What are the u- and v-values of Xnext, again rounding as above? What's p at Xnext? What does VP|xnext tell you regarding the next gradient descent step(s)? Note that it's not asked here to do any subsequent step(s).

Assume a bivariate patch p(u, v) over the unit square [0, 1]² that is given as a tensor product patch where u-sections (u fixed to some constant û; v varying across [0, 1]) are quadratic polynomials Pu:û(v) = p(û, v) while v-sections are lines pv:ô (u) = p(u, v). The boundary lines pv:o(u) and pv:1 (u) are specified by their end points p(0,0) 0.8 and p(1,0) 0.2 as well as p(0, 1) 0.3 and p(1, 1) = 0.8. The boundary quadratics pu:o(v) and pu:1 (v) interpolate p(0,0.5) = 0.1 and p(1, 0.5) = 0.9 in addition to the above given four corner-values. = = = Use Pu:û(v) = (1, v, v² ) Mq (Pu:û(0), Pu:û (0.5), Pu:û(1)) with Ma = 1 0 0 -3 4-1 2 4 2 (Pv:ô as well as pu: (u) = (1, u) M₁ (pv:v (0), P: (1)) with M₁ = = (19) 0 to formulate p(u, v) using the "geometric input" G with G = = (P(0,0%) p(0,0) p(0,0.5) p(0,1) ) = ( 0.39 0.8 0.1 0.3 0.2 0.9 0.8 p(1,0) p(1, 0.5) p(1, 1) See the figure below for (left) a selection of iso-lines of p(u, v) and (right) a 3D rendering of p(u, v) as a height surface over [0, 1]2. Black lines correspond p-levels of "/10 with n ≥ {1, 2, ………‚ 9}. 1.00 0.95 0.90 0.85 0.80 0.75 -0.15 0.70- 0.65 0.60 0.55 0.50 0.45- 0.40 0.35 15 0.30- 0.25 0.25 -0.35 Contoured patch 0.65 -0.85 0.75 0.65 0.9 0.95 0.90 0.7 0.85 0.80 0.75 0.70 0.6 0.65 0.60 0.55 0.5 p(u, v) 0.50 0.45 0.40 0.35 0.4 0.30 0.25 Patch as height over 2D 2.65° 70° 75° 80° 85°.209.85.00 0.00 0.15 0.3 0.10 0.05 0.00 1.000.95 0.90 0.850.80, 0.10 0.00 0.00 0.25 0.35 -0.55 0.10 0.6 0.55 0.5 0.20 0.45 0.45 0.15 0.5 0.6 0.55 0.5 0.5 0.45 -0.45 0.4 0.1 0.55 0.35 0.05 0.65 0.7 0.3 0.75 0.00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 -0.25- 0.95 1.00 0 น What's the "five factors formula" for p(u, v) and what's the according formula for Vp|u,v? Verify, by evaluating p(0, 1) as well as p(1, 0), that you indeed get the right values as given above “back out”, i.e., 0.3 and 0.2, respectively. Show your according calculations. = (Uinit, Vinit) T T = - Xinit Next, consider domain location Xinit (1, 0.5). What's p(xinit)? What's VP|xinit? Do one gradient descent step according to ✗next = Xinit TVPx, including a line-search for the best-possible 7. What 7-value reduces p the most? You can round this 7-value to 3 digits after the comma. What are the u- and v-values of Xnext, again rounding as above? What's p at Xnext? What does VP|xnext tell you regarding the next gradient descent step(s)? Note that it's not asked here to do any subsequent step(s).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.4: The Dot Product

Problem 10E

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