Why Check a Converted Surface?
A surface equation can look plausible after changing coordinates and still describe the wrong set. A square may have disappeared, an angle may be measured from the wrong direction, or an algebraic simplification may have removed a branch. Rather than trusting the appearance of a transformed equation, test it against cross-sections whose geometry is easy to determine in the original coordinates.
These cross-sections are called traces. They reduce a three-dimensional surface question to a two-dimensional curve or a one-dimensional slice. A trace that disagrees with the original surface proves that the conversion is wrong. Agreement on a few traces is useful evidence, but it does not by itself prove the equations equivalent.
Coordinate Conventions for the Checks
In cylindrical coordinates, \(x=r\cos\theta\), \(y=r\sin\theta\), and \(z=z\), with \(r\geq 0\). Thus \(x^2+y^2=r^2\). In spherical coordinates, we use \(\rho\geq0\), azimuthal angle \(\theta\), and polar angle \(\phi\), where \(0\leq\phi\leq\pi\):
In particular, \(x^2+y^2=\rho^2\sin^2\phi\). The angle \(\phi\) is measured from the positive \(z\)-axis. These formulas are enough to test a proposed equation; the goal here is not to derive a conversion from scratch but to check that its surface has been preserved.
A Reliable Trace-Checking Procedure
Choose a slice that makes the original equation simple. Substitute the fixed coordinate into the original equation and into the proposed converted equation, then compare the resulting traces. Coordinate planes and axes are often effective because their intersections with familiar surfaces are easy to recognize.
Try a coordinate plane such as \(z=0\) or \(x=0\), or inspect an axis. Prefer a slice with a simple, recognizable intersection.
Fix the chosen coordinate in the original Cartesian equation and simplify. Record any points, radii, or branches that remain.
Apply the same slice to the converted equation, taking account of the coordinate ranges and any exceptional cases.
If the traces differ, the conversion is incorrect. Also substitute representative points from each branch into both descriptions.
Worked Checks
Worked Example: A Missing Square in a Spherical Equation
Check the proposed cylindrical conversion of the sphere \(x^2+y^2+z^2=9\). The proposed equation is \(r+z^2=9\).
First take the horizontal trace \(z=0\). In the original equation,
This is a circle of radius \(3\). In cylindrical coordinates, \(x^2+y^2=r^2\), so the correct trace must be \(r^2=9\), which gives \(r=3\) because \(r\geq0\).
But the proposed equation gives
It predicts a circle of radius \(9\), so the trace disagrees. The error is the missing square on \(r\). The correct conversion is
As a point check, \((3,0,0)\) lies on the original sphere because \(3^2+0^2+0^2=9\). Its cylindrical coordinates have \(r=3\), \(z=0\), and satisfy \(r^2+z^2=3^2+0^2=9\). The incorrect equation gives \(r+z^2=3+0=3\), not \(9\). Both the trace and point check confirm the correction.
Worked Example: Catching an Incorrect Spherical Angle
Convert the plane \(z=0\) using the stated spherical convention, and check the result using a point and a coordinate trace.
Since \(z=\rho\cos\phi\), the plane equation becomes \(\rho\cos\phi=0\). For \(\rho>0\), this requires \(\cos\phi=0\). In the range \(0\leq\phi\leq\pi\), the solution is \(\phi=\pi/2\). The origin, where \(\rho=0\), also lies in the plane. Thus the plane is represented by \(\phi=\pi/2\), together with the origin (which is already included by \(\rho=0\) for any angles).
Check the point \((1,0,0)\). Its spherical coordinates have \(\rho=1\), \(\theta=0\), and \(\phi=\pi/2\), since
So the point is on both the plane and the converted description.
A common angle-convention error is to write \(\phi=0\) for the plane \(z=0\). But \(\phi=0\) gives \(x=0\), \(y=0\), and \(z=\rho\): it describes the positive \(z\)-axis, not the equatorial plane. For example, \((1,0,0)\) is on \(z=0\), but it cannot be described by \(\phi=0\), because that angle forces \(x=y=0\). The correct value \(\phi=\pi/2\) is confirmed by the convention and by the point check.
A related trace check is the intersection with the \(xz\)-plane, where \(y=0\). The original plane \(z=0\) meets it in the \(x\)-axis. In spherical coordinates, the points with \(\phi=\pi/2\) and \(\theta=0\) or \(\theta=\pi\) have \(z=0\), \(y=0\), and \(x=\rho\) or \(x=-\rho\), respectively. Both halves of the \(x\)-axis are present.
Worked Example: Detecting a Lost Branch of a Cone
Check the spherical description of the double cone \(z^2=x^2+y^2\). Substitution gives
If \(\rho>0\), divide by \(\rho^2\) to obtain \(\cos^2\phi=\sin^2\phi\). On \(0\leq\phi\leq\pi\), this yields \(\phi=\pi/4\) or \(\phi=3\pi/4\). The angle \(\theta\) is unrestricted. The origin, where \(\rho=0\), also satisfies the original equation.
Check the vertical trace \(y=0\). In Cartesian coordinates, the trace equation is
This consists of two lines in the \(xz\)-plane. In spherical coordinates, \(y=0\) occurs for \(\theta=0\) or \(\theta=\pi\) (for points away from the axis). The angle \(\phi=\pi/4\) gives \(z=\rho/\sqrt{2}\) and, at \(\theta=0\), \(x=\rho/\sqrt{2}\), so \(z=x\). At \(\theta=\pi\), \(x=-\rho/\sqrt{2}\), so \(z=-x\). The angle \(\phi=3\pi/4\) gives \(z=-\rho/\sqrt{2}\); it supplies the other halves of those lines. The trace therefore contains both cone branches.
If one reports only \(\phi=\pi/4\), the lower nappe is lost. The point \((0,0,-2)\) shows the failure directly: it satisfies the original equation because \((-2)^2=0^2+0^2=4\). Its spherical coordinates have \(\rho=2\), \(\phi=\pi\), and any \(\theta\), so it is not represented by \(\phi=\pi/4\). Dividing by \(\rho^2\) was valid only for \(\rho>0\), and solving the resulting equation requires both angles in the allowed range. The origin must also be retained as a solution of the original equation.
Why Traces Are Useful—and Their Limit
There is a simple reason a trace comparison is a sound diagnostic. If two surface descriptions represent the same set \(S\), then intersecting both with the same plane \(P\) gives the same set \(S\cap P\). Therefore, unequal traces prove that the surface descriptions are not equivalent. This conclusion follows directly from the definition of intersection: a point belongs to \(S\cap P\) exactly when it belongs to both \(S\) and \(P\).
The converse does not follow from checking only a few traces. For example, the planes \(z=0\) and \(z=x\) have the same trace in the plane \(x=0\): both intersections are \(x=0,\ z=0\), with \(y\) free. Yet the two planes differ; the point \((1,0,0)\) lies on \(z=0\) but not on \(z=x\). Thus a successful trace check can increase confidence and reveal errors, but a complete verification needs an algebraic equivalence argument, with all coordinate restrictions and exceptional cases accounted for.
Common Pitfalls
- Forgetting a square. Since \(x^2+y^2=r^2\), a cylindrical radial term may not be replaced by \(r\). Compare radii in a horizontal trace.
- Using a different spherical angle convention. Here \(z=\rho\cos\phi\), so \(\phi=0\) points along the positive \(z\)-axis and \(\phi=\pi/2\) is the equatorial plane. Test a point on the plane or axis.
- Dividing without checking the zero case. If an equation has a factor \(\rho\) or \(\rho^2\), treat \(\rho=0\) separately before dividing.
- Keeping only one solution branch. Squared trigonometric equations can have multiple angles in the permitted range. Check a point on each branch of the original surface.
- Treating a few matching traces as proof. Traces are necessary consistency checks, not a substitute for showing that the complete descriptions are equivalent.
Check Your Understanding
For each question, explain what the proposed check can establish.
- What is the trace of a surface in the plane \(z=0\), and why does a mismatch with a proposed conversion show that the conversion is wrong?
- The sphere \(x^2+y^2+z^2=16\) is converted to \(r+z^2=16\). What does the trace \(z=0\) show about this equation?
- Under the spherical convention in this tutorial, what angle \(\phi\) describes the plane \(z=0\) away from the origin? What does \(\phi=0\) describe?
- Why must the origin be checked separately when dividing a spherical equation by \(\rho^2\)?
- Can agreement on one coordinate-plane trace prove that two surfaces are identical? Give a reason for your answer.