Equation of a section plane in hyperbolic paraboloidFind the equation of a plane passing through Ox and intersecting a hyperbolic paraboloid x2p−y2q=2z(p&gt;0,q&gt;0) along a hyperbola with equal semi-axes.My attempt: The equation of a plane passing through Ox is By+Cz=0. So z=−ByC. Now substitute z=−ByC into the equation of hyperbolic paraboloid x2p−y2q=−2ByC and transform to x2p−(y−BqC)2q=−B2qC2. What to do next? I don't understand how to find B and C if we assume x2p−(y−BqC)2q=−B2qC2 is a hyperbola with equal semi-axes.

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Equation of a section plane in hyperbolic paraboloidFind the equation of a plane passing through Ox and intersecting a hyperbolic paraboloid x2p−y2q=2z(p&amp;gt;0,q&amp;gt;0) along a hyperbola with equal semi-axes.My attempt: The equation of a plane passing through Ox is By+Cz=0. So z=−ByC. Now substitute z=−ByC into the equation of hyperbolic paraboloid x2p−y2q=−2ByC and transform to x2p−(y−BqC)2q=−B2qC2. What to do next? I don&#039;t understand how to find B and C if we assume x2p−(y−BqC)2q=−B2qC2 is a hyperbola with equal semi-axes.

arrebusnaavbz · Accepted Answer

A generic plane containing the x axis (Ox) has a unit normal vectorn=(0,cos⁡θ,sin⁡θ)Two vectors (unit and mutually perpendicular) that span this plane arev1=(1,0,0) and v2=(0,sin⁡θ,−cos⁡θ)Hence, on this plane, the position vector of any point is r=[v1,v2]u where u=[u1,u2]T is the coordinate vector of r with respect to v1,v2.Hence, x=u1y=sin⁡θu2z=−cos⁡θu2Plug this into the equation of the hyperbolic paraboloid,du12p−dsin2θu22q=−2cos⁡θu2Swhich is an equation of hyperbola in u1 and u2. If the semi-axes are equal thenp=dqsin2θHence, the required angle θ satisfies |sin⁡θ|=dqpThis determines four possible values for θ and correspondingly four planes.

kachnaemra · Accepted Answer

The last equation you got is that of a hyperbola, which is the projection of the hyperboloid section onto xy plane. The projection has semi-axes p (parallel to x) and q (parallel to y).The tilted plane forms an angle α (with tan⁡α=−BC) with xy plane, and is parallel to x axis. Hence the hyperbola on the plane has semi-axes p (parallel to x) and q|sin⁡α| (parallel to y). You then get a right hyperbola if |sin⁡α|=qp.

Equation of a section plane in hyperbolic paraboloid Find

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