Comparing Quadrics Through Horizontal Cross-Sections

Reading a Surface One Height at a Time

A three-dimensional surface can be difficult to recognize from its equation alone. One useful strategy is to examine what remains when the surface is cut by horizontal planes. Each plane has a fixed height, so its intersection with the surface is a two-dimensional curve. As the height changes, the curves reveal how the surface is organized.

For ellipsoids, cones, and hyperboloids whose axes are parallel to the \(z\)-axis, these horizontal intersections are typically ellipses, with a point possible at a tip or boundary. The key is not just to notice that the slices are elliptical. Track whether the ellipses shrink, grow, or appear only above or below a certain height.

How to Read the Ellipse in a Slice

An equation of the form

describes an ellipse in the horizontal plane, provided \(A^2>0\) and \(B^2>0\). Its semiaxis lengths are \(A\) in the \(x\)-direction and \(B\) in the \(y\)-direction, where \(A\) and \(B\) are positive. If the right-hand side is a positive number \(q\), then

Thus the semiaxis lengths are \(a\sqrt{q}\) and \(b\sqrt{q}\). The same scale factor \(\sqrt{q}\) changes both semiaxes. If \(q=0\), the equation reduces to \(x=y=0\), a single point. If \(q<0\), there are no real points because a sum of nonnegative terms cannot be negative.

This gives a consistent way to compare surfaces: at each height, find the factor on the right-hand side of the normalized ellipse equation. Then determine whether that factor is positive, zero, or negative and how it changes with height.

Ellipsoids: Largest at the Middle

Consider an ellipsoid centered at the origin with its principal axis along \(z\):

At height \(z=k\), substitution gives

The right-hand side is positive only when \(-c<k<c\). At \(k=0\), the horizontal ellipse has semiaxes \(a\) and \(b\), so it is the largest horizontal slice. As \(|k|\) increases toward \(c\), the right-hand side decreases toward zero and both semiaxes shrink. At \(k=c\) or \(k=-c\), the slice is a point; for \(|k|>c\), there is no intersection.

Cones: A Point and Two Expanding Directions

A double elliptic cone with vertex at the origin and axis along \(z\) can be written as

At height \(z=k\), the slice is

For \(k\ne 0\), this is an ellipse whose semiaxes are \(a|k|/c\) and \(b|k|/c\). At \(k=0\), it is just the vertex. The slices grow as \(|k|\) grows, both above and below the vertex. Unlike an ellipsoid, a double cone does not end at a largest height: its slices continue to expand as the distance from the vertex increases.

Hyperboloids: Two Different Slice Patterns

A hyperboloid of one sheet with axis along \(z\) has the standard form

At height \(k\), its horizontal slice satisfies

The right-hand side is always positive. Every height has an elliptical cross-section. At \(k=0\), the ellipse has semiaxes \(a\) and \(b\); as \(|k|\) increases, the right-hand side increases and the slices grow. So the surface has a smallest horizontal slice at its middle, then widens in both directions.

A hyperboloid of two sheets with its axis along \(z\) is written

At height \(k\), rearranging gives

A real ellipse exists only when \(|k|>c\). At \(k=c\) or \(k=-c\), the section is a point, and for \(|k|<c\), it is empty. Beyond either boundary height, the ellipses grow as \(|k|\) increases. The empty band between the two sheets is the crucial feature distinguishing this surface from a one-sheet hyperboloid.

Why Substitution Gives the Cross-Section

The method is direct because a horizontal plane at height \(k\) consists exactly of the points whose \(z\)-coordinate equals \(k\). To find points belonging to both that plane and the surface, impose \(z=k\) in the surface equation. The remaining equation describes precisely the allowed \(x\)- and \(y\)-coordinates at that height.

For example, applying this reasoning to a one-sheet hyperboloid gives

Moving the fixed height term to the other side yields

The right-hand side is positive at every real height \(k\), so every horizontal plane intersects the surface. Since \(k^2\) increases with \(|k|\), the right side—and therefore the squared scale of the ellipse—increases as one moves away from the central slice. This derives both the existence of every slice and its growth pattern.

A Reliable Identification Procedure

When the equation or a sequence of slices is unfamiliar, separate the evidence into three questions: What shape is each nonempty section? At which heights does it exist? How does its size change with height?

The patterns can be summarized as follows.

A Common Sign Mistake

For a one-sheet hyperboloid, the negative \(z^2\) term must be moved carefully. Starting with

substituting \(z=k\) gives

A common incorrect step is to write the right side as \(1-k^2/c^2\). That is the wrong sign: adding \(k^2/c^2\) to both sides gives \(1+k^2/c^2\), not \(1-k^2/c^2\). The incorrect expression would imply that the sections disappear when \(|k|>c\), which contradicts the original equation. For instance, at \(k=2c\), the correct slice has right side \(1+4=5\), while the incorrect one gives \(-3\), an impossible ellipse. Substituting \(z=2c\) into the original equation confirms the correct relation: the \(x,y\) terms must sum to \(1+4=5\).