Boundary Curves Of Graph And Standard Patches

How Parameter Ranges Trace a Surface Boundary

A parametrized surface is described by two parameters. When their allowed values fill a rectangle, the rectangle has four edges: two where the first parameter is fixed and two where the second is fixed. Mapping those edges onto the surface gives the curves along the patch boundary.

This simple procedure applies to graph patches and to standard spherical and cylindrical patches. The key is to change only one parameter at a time. Fixing both parameters gives a corner point, not an entire boundary curve.

Definition: Let \(X(u,v)\) parametrize a surface patch for \(a\le u\le b\) and \(c\le v\le d\). The four parameter-edge curves are \(X(a,v)\), \(X(b,v)\), \(X(u,c)\), and \(X(u,d)\), with the unfixed parameter varying over its full interval. Their images are the candidate boundary curves of the patch.

“Candidate” matters. If the parametrization is one-to-one on the rectangle and gives a regular patch, these curves describe its boundary. If the parametrization repeats points, has a seam, or collapses an edge to a point, some curves may coincide or may not represent distinct geometric edges.

Why Restricting a Parameter Finds the Boundary

Begin with the parameter rectangle \(R=[a,b]\times[c,d]\). Its boundary consists of the four sides \(u=a\), \(u=b\), \(v=c\), and \(v=d\). A parametrization \(X\) sends each point of this rectangle to a point on the surface, so the images of those four sides are exactly the four restricted curves.

For a one-to-one parametrization that is a homeomorphism from the closed rectangle onto the patch, boundary points of the patch correspond to boundary points of the rectangle. Consequently, the patch boundary is the image of the rectangle boundary. This is the reason the restriction method works; it is not a rule for guessing curves from a sketch.

\partial R = \{(a,v):c\le v\le d\} \cup \{(b,v):c\le v\le d\} \cup \{(u,c):a\le u\le b\} \cup \{(u,d):a\le u\le b\}.

Applying \(X\) to these four sets gives the four restricted curves. In a non-injective parametrization, this calculation still identifies the images of the parameter edges, but the images must be checked: two sides may map to the same curve, or a whole side may map to one point.

Graph Patches

A graph patch is parametrized by allowing the two independent coordinates to vary over a region \(D\) in the \(xy\)-plane. For a graph \(z=f(x,y)\), a standard parametrization is \(X(x,y)=(x,y,f(x,y))\). If \(D\) is a rectangle, fixing \(x\) or \(y\) at either endpoint traces one of the four edges. The height on each curve is found by substituting the fixed and varying coordinates into \(f\).

Worked Example: Boundary of a Graph Patch

Find the boundary curves of the graph \(z=x^2+2y\) over the rectangle \(-1\le x\le1\), \(0\le y\le2\).

Use the graph parametrization \(X(x,y)=(x,y,x^2+2y)\). First fix \(x=-1\) and let \(y\) vary:

C_1(y)=X(-1,y)=((-1),y,(-1)^2+2y)=(-1,y,1+2y), \qquad 0\le y\le2.

At the other \(x\)-endpoint, \(x=1\):

C_2(y)=X(1,y)=(1,y,1^2+2y)=(1,y,1+2y), \qquad 0\le y\le2.

Now fix \(y=0\) and let \(x\) vary:

C_3(x)=X(x,0)=(x,0,x^2+2(0))=(x,0,x^2), \qquad -1\le x\le1.

Finally, fix \(y=2\):

C_4(x)=X(x,2)=(x,2,x^2+2(2))=(x,2,x^2+4), \qquad -1\le x\le1.

Check the corners. At \((x,y)=(-1,0)\), the surface point is \((-1,0,1)\), which is both \(C_1(0)\) and \(C_3(-1)\). At \((1,0)\), it is \((1,0,1)\), both \(C_2(0)\) and \(C_3(1)\). At \((-1,2)\), it is \((-1,2,5)\), both \(C_1(2)\) and \(C_4(-1)\). At \((1,2)\), it is \((1,2,5)\), both \(C_2(2)\) and \(C_4(1)\). Each neighboring pair of curves therefore meets at the appropriate corner.

Spherical Sectors

For a sphere of radius \(R\), use the standard spherical parametrization with polar angle \(\phi\) measured from the positive \(z\)-axis and azimuthal angle \(\theta\) in the \(xy\)-plane:

X(\theta,\phi) = (R\sin\phi\cos\theta,\ R\sin\phi\sin\theta,\ R\cos\phi).

On a rectangular parameter range, fixing \(\theta\) traces a meridian curve, while fixing \(\phi\) traces a latitude curve. At \(\phi=0\) or \(\phi=\pi\), the longitude-dependent terms vanish and a constant-\(\phi\) edge collapses to a pole. If the azimuthal range covers a full revolution, its two endpoint meridians coincide. These are reasons to inspect the restricted curves rather than assume all four images are distinct.

Worked Example: Boundary of a Spherical Sector

Take the sphere of radius \(3\), with \(0\le\theta\le\frac{\pi}{2}\) and \(\frac{\pi}{3}\le\phi\le\frac{2\pi}{3}\). Find its boundary curves by restricting each parameter to its endpoints.

Here \(R=3\), \(\sin(\frac{\pi}{3})=\sin(\frac{2\pi}{3})=\frac{\sqrt3}{2}\), \(\cos(\frac{\pi}{3})=\frac12\), and \(\cos(\frac{2\pi}{3})=-\frac12\). Fix \(\theta=0\):

C_1(\phi)=X(0,\phi) =(3\sin\phi,0,3\cos\phi), \qquad \frac{\pi}{3}\le\phi\le\frac{2\pi}{3}.

Fix \(\theta=\frac{\pi}{2}\), using \(\cos(\frac{\pi}{2})=0\) and \(\sin(\frac{\pi}{2})=1\):

C_2(\phi)=X\left(\frac{\pi}{2},\phi\right) =(0,3\sin\phi,3\cos\phi), \qquad \frac{\pi}{3}\le\phi\le\frac{2\pi}{3}.

Fix \(\phi=\frac{\pi}{3}\):

C_3(\theta)=X\left(\theta,\frac{\pi}{3}\right) =\left(\frac{3\sqrt3}{2}\cos\theta, \frac{3\sqrt3}{2}\sin\theta, \frac32\right), \qquad 0\le\theta\le\frac{\pi}{2}.

Fix \(\phi=\frac{2\pi}{3}\):

C_4(\theta)=X\left(\theta,\frac{2\pi}{3}\right) =\left(\frac{3\sqrt3}{2}\cos\theta, \frac{3\sqrt3}{2}\sin\theta, -\frac32\right), \qquad 0\le\theta\le\frac{\pi}{2}.

For an endpoint check, \(C_1(\frac{\pi}{3})=(\frac{3\sqrt3}{2},0,\frac32)\), which equals \(C_3(0)\). Also \(C_2(\frac{\pi}{3})=(0,\frac{3\sqrt3}{2},\frac32)=C_3(\frac{\pi}{2})\). At the other latitude, \(C_1(\frac{2\pi}{3})=(\frac{3\sqrt3}{2},0,-\frac32)=C_4(0)\), and \(C_2(\frac{2\pi}{3})=(0,\frac{3\sqrt3}{2},-\frac32)=C_4(\frac{\pi}{2})\). The four restricted curves meet at the four parameter corners.

Cylindrical Segments

A circular cylinder with its axis along the \(z\)-axis can be parametrized by \(X(\theta,z)=(r\cos\theta,r\sin\theta,z)\), where \(r\) is its radius. Fixing \(z\) traces a circular rim; fixing \(\theta\) traces a straight generator along the cylinder. A finite lateral segment has two rim curves and two parameter-edge curves that need special attention: if \(\theta\) spans a full revolution, the two generator curves at the azimuthal endpoints are the same seam, not separate geometric edges.

Worked Example: Boundary of a Quarter-Cylinder Segment

Find the boundary curves of the lateral patch of radius \(2\), with \(0\le\theta\le\frac{\pi}{2}\) and \(-1\le z\le3\). Use \(X(\theta,z)=(2\cos\theta,2\sin\theta,z)\).

Fix \(\theta=0\) and let \(z\) vary:

C_1(z)=X(0,z)=(2\cos0,2\sin0,z)=(2,0,z), \qquad -1\le z\le3.

Fix \(\theta=\frac{\pi}{2}\):

C_2(z)=X\left(\frac{\pi}{2},z\right) =(2\cos\frac{\pi}{2},2\sin\frac{\pi}{2},z) =(0,2,z), \qquad -1\le z\le3.

Fix \(z=-1\):

C_3(\theta)=X(\theta,-1)=(2\cos\theta,2\sin\theta,-1), \qquad 0\le\theta\le\frac{\pi}{2}.

Fix \(z=3\):

C_4(\theta)=X(\theta,3)=(2\cos\theta,2\sin\theta,3), \qquad 0\le\theta\le\frac{\pi}{2}.

Check the shared endpoints: \(C_1(-1)=(2,0,-1)=C_3(0)\), \(C_2(-1)=(0,2,-1)=C_3(\frac{\pi}{2})\), \(C_1(3)=(2,0,3)=C_4(0)\), and \(C_2(3)=(0,2,3)=C_4(\frac{\pi}{2})\). These four curves make the boundary of this quarter-cylinder patch.

A Common Mistake: Fixing Both Parameters

A frequent error is to set both parameters to endpoints and report the resulting points as the boundary. For the graph example, setting \(x=-1\) and \(y=0\) gives only \(X(-1,0)=(-1,0,1)\), one corner. It does not give the entire edge \(C_1(y)=(-1,y,1+2y)\). The correction is to fix one parameter at an endpoint while allowing the other to range across its full interval.

Another useful check is to evaluate the endpoints of every restricted curve. In a rectangular parameter patch, adjacent sides must agree at their shared corner. If they do not, a parameter may have been substituted incorrectly, or the parameter intervals may have been recorded incorrectly.

Takeaway: For each endpoint of each parameter range, hold that parameter fixed and let the other vary. Then check for repeated seams, collapsed edges, and agreement at the corners.

Check Your Understanding

For each question, identify which parameter should be fixed and what the remaining parameter range is.

For \(X(u,v)=(u,v,u^2-v)\) on \([-2,1]\times[0,3]\), what four restrictions give the candidate boundary curves?
On a spherical patch, what kind of curve results when the polar angle is fixed? What kind results when the azimuthal angle is fixed?
Why can a constant-polar-angle boundary curve collapse to a point at a pole?
For a full circular cylinder, why do the curves at the two azimuthal endpoints represent the same seam?
What corner check can detect a mistake in one of the restricted curves?