Introduction
A surface parametrization is a map from a region in the parameter plane into three-dimensional space. Its formulas and bounds work together: the formulas determine where each parameter pair goes, while the bounds determine which part of the surface is reached and how often it is reached.
A plausible-looking formula can therefore describe the wrong patch. Bounds may leave out part of the surface or trace the same points repeatedly. Even when the image is correct, the parameter tangent vectors may become dependent, or their cross product may point opposite to the required normal direction.
The most reliable way to assess a parametrization is to check these issues separately. First identify the intended surface patch; then compare its points with the image of the proposed map, test regularity, and check orientation if a normal direction is specified.
A Practical Diagnostic
Do not treat correctness as one yes-or-no property. A parametrization can have the right formula but wrong bounds, the right image but an unwanted repeated trace, or the right image and coverage but a singular point or reversed normal. For a proposed parametrization, ask:
- What points does the target patch contain?
- Do the parameter bounds reach every target point?
- Do distinct parameter pairs represent the same point, and if so, is that only an expected boundary identification?
- Are the two parameter tangent vectors independent at the points where regularity is required?
- If an orientation is specified, does \(\mathbf{r}_u\times\mathbf{r}_v\) point in the required direction?
These checks concern different features. In particular, having the correct image does not by itself guarantee regularity or the desired orientation.
Worked Example: Bounds That Omit Part of a Plane Patch
Worked Example: Bounds That Omit Part of a Plane Patch
Consider the rectangle in the plane \(z=0\) specified by \(0\le x\le2\) and \(0\le y\le1\). A proposed parametrization is \(\mathbf{r}(u,v)=(u,v,0)\), with \(0\le u\le1\) and \(0\le v\le1\). Does it cover the specified rectangle?
Substituting the bounds into the formula gives \(x=u\) and \(y=v\), so the image satisfies \(0\le x\le1\) and \(0\le y\le1\). For instance, the target point \((3/2,1/2,0)\) is not reached: it would require \(u=3/2\), outside the stated domain.
Keep the same formula and correct the bounds to \(0\le u\le2\), \(0\le v\le1\). For any point \((x,y,0)\) in the target rectangle, choose \(u=x\) and \(v=y\). Those values lie in the corrected domain and give \(\mathbf{r}(x,y)=(x,y,0)\). Conversely, every permitted pair gives a point satisfying the target inequalities.
Regularity is checked directly: \(\mathbf{r}_u=(1,0,0)\) and \(\mathbf{r}_v=(0,1,0)\), hence \(\mathbf{r}_u\times\mathbf{r}_v=(0,0,1)\ne\mathbf{0}\). The corrected parametrization covers the whole rectangle and is regular.
This example illustrates a useful coverage check: show both that every parametrized point lies in the target patch and that every target point can be represented by allowed parameters. The first direction alone cannot detect an omitted strip.
Worked Example: Repeated Coverage of a Cylinder
Worked Example: Repeated Coverage of a Cylinder
Parametrize the lateral surface of the unit cylinder \(x^2+y^2=1\) for \(0\le z\le1\). A proposed map is \(\mathbf{r}(\theta,z)=(\cos\theta,\sin\theta,z)\), but the bounds are \(0\le\theta\le4\pi\), \(0\le z\le1\).
Every image point satisfies \(x^2+y^2=\cos^2\theta+\sin^2\theta=1\), and its height is in the required interval. However, increasing \(\theta\) by \(2\pi\) leaves both trigonometric coordinates unchanged. Thus, for a typical point with angle \(\theta\) strictly between \(0\) and \(2\pi\), both \((\theta,z)\) and \((\theta+2\pi,z)\) are in the proposed domain and map to the same point. The cylinder is traced twice.
Use \(0\le\theta\le2\pi\) and \(0\le z\le1\) instead. Any point \((x,y,z)\) on the target cylinder has a height in \([0,1]\), and its unit-circle coordinates can be written \((\cos\theta,\sin\theta)\) for an angle in \([0,2\pi]\). So the corrected bounds reach every point. The two endpoint angles \(0\) and \(2\pi\) describe the same seam; this is the expected boundary identification for this closed parameter interval, not a second full traversal of the cylinder.
For regularity and orientation, compute the tangent vectors: \(\mathbf{r}_\theta=(-\sin\theta,\cos\theta,0)\) and \(\mathbf{r}_z=(0,0,1)\). Their cross product is \(\mathbf{r}_\theta\times\mathbf{r}_z=(\cos\theta,\sin\theta,0)\). It is nonzero and points radially outward. This verifies regularity and gives the outward normal direction.
Repeated coverage is not the same error as incomplete coverage. Restricting an angle interval too far can omit points; making it too long can repeat points. Periodic coordinates require particular care because distinct parameter values can have identical images.
Worked Example: A Spherical Cap and a Coordinate Singularity
Worked Example: A Spherical Cap and a Coordinate Singularity
Consider the part of the unit sphere \(x^2+y^2+z^2=1\) above the plane \(z=1/2\). In spherical coordinates, let \(\mathbf{r}(\phi,\theta)=(\sin\phi\cos\theta,\sin\phi\sin\theta,\cos\phi)\), where \(\phi\) is the angle from the positive \(z\)-axis.
The condition \(z\ge1/2\) becomes \(\cos\phi\ge1/2\), so the cap requires \(0\le\phi\le\pi/3\). Full rotation around the axis requires \(0\le\theta\le2\pi\). Choosing \(\pi/3\le\phi\le\pi/2\), for example, would produce a band below the cap’s boundary rather than the stated cap.
There is a further issue at the north pole. Differentiation gives \(\mathbf{r}_\phi=(\cos\phi\cos\theta,\cos\phi\sin\theta,-\sin\phi)\) and \(\mathbf{r}_\theta=(-\sin\phi\sin\theta,\sin\phi\cos\theta,0)\). At \(\phi=0\), \(\mathbf{r}_\theta=(0,0,0)\), so the parametrization is not regular there. Every value of \(\theta\) at \(\phi=0\) represents the same pole. The image covers the cap, but this spherical-coordinate chart is singular at a point in the cap.
For a regular parametrization of this cap, use \(x\) and \(y\) as parameters: \(\mathbf{q}(x,y)=(x,y,\sqrt{1-x^2-y^2})\), with \(x^2+y^2\le3/4\). Indeed, \(z\ge1/2\) on the cap implies \(x^2+y^2=1-z^2\le3/4\). Conversely, these disk bounds make \(z=\sqrt{1-x^2-y^2}\ge1/2\), so the map reaches exactly the cap, using the positive square root.
Writing \(z=\sqrt{1-x^2-y^2}\), the tangent vectors are \(\mathbf{q}_x=(1,0,-x/z)\) and \(\mathbf{q}_y=(0,1,-y/z)\). Therefore \(\mathbf{q}_x\times\mathbf{q}_y=(x/z,y/z,1)\), whose third component is \(1\). The cross product is never zero on this domain, so this parametrization is regular, including at the pole \((x,y)=(0,0)\).
The check distinguishes an error in the patch bounds from a singularity in the chosen coordinates. Correcting the angle bounds makes the spherical parametrization cover the desired cap, but it does not remove its pole singularity.
Worked Example: A Reversed Normal
Worked Example: A Reversed Normal
Parametrize the plane rectangle \(0\le x\le1\), \(0\le y\le2\), \(z=0\), with the upward orientation. The map \(\mathbf{r}(u,v)=(u,v,0)\), on \(0\le u\le1\), \(0\le v\le2\), has \(\mathbf{r}_u=(1,0,0)\) and \(\mathbf{r}_v=(0,1,0)\). Thus \(\mathbf{r}_u\times\mathbf{r}_v=(0,0,1)\), which points upward as required.
If the parameters are swapped, \(\widetilde{\mathbf{r}}(u,v)=(v,u,0)\), with \(0\le u\le2\), \(0\le v\le1\), the image is still exactly the same rectangle. But \(\widetilde{\mathbf{r}}_u=(0,1,0)\) and \(\widetilde{\mathbf{r}}_v=(1,0,0)\), so \(\widetilde{\mathbf{r}}_u\times\widetilde{\mathbf{r}}_v=(0,0,-1)\). This is downward, opposite to the required orientation.
To correct the orientation, reverse the parameter order in the cross product: \(\widetilde{\mathbf{r}}_v\times\widetilde{\mathbf{r}}_u=(0,0,1)\). Equivalently, use the original parametrization with its original parameter order. In either case, the image remains the same and the chosen normal points upward.
Why the Regularity Test Works
The regularity test is a direct translation of what it means for the surface to have two independent tangent directions. At a parameter pair \((u,v)\), small changes in \(u\) and \(v\) produce tangent vectors \(\mathbf{r}_u\) and \(\mathbf{r}_v\). If these vectors are linearly dependent, they do not span two independent directions, and their cross product is zero. If they are linearly independent, their cross product is nonzero and perpendicular to both.
In matrix terms, the derivative of \(\mathbf{r}\) has the two tangent vectors as its columns. Its rank is \(2\) precisely when those columns are independent. Thus the following conditions are equivalent at an interior parameter pair:
The cross product also explains orientation. It is perpendicular to the two tangent directions, and reversing their order changes its sign: \(\mathbf{r}_v\times\mathbf{r}_u=-(\mathbf{r}_u\times\mathbf{r}_v)\). Therefore, swapping parameter order preserves the image but reverses the normal direction. A parametrization should not be judged by its image alone when an orientation is part of the task.
Common Mistakes and Corrections
A frequent mistake is to check only that the formula produces points on the surface. In the plane example, every point produced by the too-small domain was indeed on the target rectangle, but the point \((3/2,1/2,0)\) was missing. The correction was to verify both containments: all parametrized points are in the target patch, and every target point comes from allowed parameter values.
Another mistake is to assume that a visually correct image guarantees regularity. The spherical parametrization reached the entire cap, but its \(\theta\)-tangent vanished at the pole. When regularity is required, compute the tangent vectors and test their cross product rather than relying on the appearance of the formula.
Finally, an outward or upward direction must be checked against the order of the parameters. A nonzero cross product proves regularity, not that the normal points the right way. If its direction is wrong, reverse the parameter order or change the parametrization consistently.
Check Your Understanding
For each question, identify the particular parametrization issue and the check that would reveal it.
- For \(\mathbf{r}(u,v)=(u,v,0)\), what points are missed if \(0\le u\le1\), \(0\le v\le1\) is used for the rectangle \(0\le x\le2\), \(0\le y\le1\)?
- Why does allowing \(0\le\theta\le4\pi\) in the standard cylinder parametrization repeat most points?
- At what parameter values does the spherical parametrization of the cap fail the regularity test, and which tangent vector vanishes there?
- If \(\mathbf{r}_u\times\mathbf{r}_v\) points downward but the required orientation is upward, what simple parameter-order change reverses the normal?
- What two containments should be checked to establish that a parametrization has exactly the intended image?