Answer to Question #97940 in Calculus for Rachel

Question #97940
Find a parametric representation g(s,t) of the part of the one-sheeted hy-perboloid x^2/a^2+y^2/a^2−z^2/c^2= 1 that lies between z=−1 and z= 1
1
Expert's answer
2019-11-04T09:39:40-0500

If in the form


"{x^2 \\over a^2}+{y^2\\over b^2}-{z^2 \\over c^2}=1"

then let "z=cv, v" as one parameter, and then, the equation can be rewritten in the following way:


"{x^2 \\over a^2}+{y^2\\over b^2}-{c^2v^2 \\over c^2}=1""{x^2 \\over a^2}+{y^2\\over b^2}=1+v^2"

Let "x=a\\sqrt{1+v^2}\\cos u, y=b\\sqrt{1+v^2}\\sin u." Then

"{(a\\sqrt{1+v^2}\\cos u)^2 \\over a^2}+{(b\\sqrt{1+v^2}\\sin u)^2\\over b^2}=1+v^2,"

"x=a\\sqrt{1+v^2}\\cos u,""y=b\\sqrt{1+v^2}\\sin u,""z=cv,""u\\in[0, 2\\pi), v\\in\\big [-{ 1\\over c} , {1 \\over c} \\big]."

Other parameterizations include


"x=a\\cosh{v}\\cos{u},"

"y=b\\cosh{v}\\sin{u},"

"z=c\\sinh{v},"

"u\\in[0, 2\\pi), v\\in\\big [-\\sinh^{-1}({ 1\\over c}) , \\sinh^{-1}({ 1\\over c})]."


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