The parametric equations in these exercises represent a quadric surface for positive values of a , b , and c. Identify the type of surface by eliminating the parameters u and v . Check your conclusion by choosing specific values for the constants and generating the surface with a graphing utility. x = a sinh v , y = b sinh u cosh v , z = c cosh u cosh v
The parametric equations in these exercises represent a quadric surface for positive values of a , b , and c. Identify the type of surface by eliminating the parameters u and v . Check your conclusion by choosing specific values for the constants and generating the surface with a graphing utility. x = a sinh v , y = b sinh u cosh v , z = c cosh u cosh v
The parametric equations in these exercises represent a quadric surface for positive values of a, b, and c. Identify the type of surface by eliminating the parameters u and v. Check your conclusion by choosing specific values for the constants and generating the surface with a graphing utility.
x
=
a
sinh
v
,
y
=
b
sinh
u
cosh
v
,
z
=
c
cosh
u
cosh
v
Need only a handwritten solution only (not a typed one).
Project: S is a surface in R° with equation x² + y² – z² = 1. The point P (2,1,2) lies on this hyperboloid. It
turns out there are exactly 2 straight lines L, and L2 which pass through the point P and lie entirely inside the
surface S.
Your job: Find the parametric equations of these 2 lines and then display the surface S together with the lines L1
and L2 using computer graphics software.
Methods: Theory: For finding the equations, I suggest this approach. (You can use a different approach if you know
a better one.) The two lines can be found if you know direction vectors for them. To get the direction vector v =
(a, b, c)for L1 or L2 write down the parametric equations of a line with direction vector v and passing through P.
The condition that the line lies on S gives an equation involving t, a, b and c. Solve this equation to find two
solutions for a,b,c.
Explain what it means for a surface to be orientable.
Precalculus: Mathematics for Calculus (Standalone Book)
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