Question:
(a) | Part of a landscape generated in a computer game is defined by the quadric surface \[z=3{{x}^{2}}+2xy+{{y}^{2}}+10x+2y+5\]. | |||
(i) | Obtain the critical point of the surface and determine if it is a maximum, a minimum or a saddle point. | |||
(ii) | Determine the equation of the tangent plane to the surface at the point where \[x=-1\] and \[y=0\]. | |||
(iii) | Find or state a vector that is parallel to the normal to the surface where \[x=-1\] and \[y=0\]. | [11] | ||
(b) | (i) | Construct the bilinear surface patch based on the line segments joining \[A\left( 1,\,1,\,0 \right)\] with \[B\left( 2,\,1,\,1 \right)\] and \[C\left( 1,\,2,\,0 \right)\] with \[D\left( 2,\,2,\,0 \right)\]. | ||
(ii) | Obtain the parametric equations of the curve joining A to D. | |||
(iii) | Show that the curve joining B to C is a parabola lying in the plane \[x+y=3\]. | [9] |