Geometry Of A Surface Of Revolution

From a Curve to a Surface

A surface of revolution is formed by rotating a plane curve around a fixed line. The curve is the profile, and the fixed line is the axis of revolution. Each point on the profile traces a circle centered on the axis. A short segment of the profile sweeps out a narrow band of the surface.

To understand the geometry of that band, we need to keep track of two distances: how far the profile point is from the axis, and how long the short profile segment is. The first determines the radius of the circle traced by the point. The second gives the local width of the swept band.

Key idea: A profile point sweeps out a circle whose radius is its perpendicular distance from the axis. A short profile segment sweeps out a narrow band whose local width is determined by the segment’s arc length.

Profile, Axis, and Perpendicular Radius

For a concrete coordinate description, take the profile to lie in the \(xy\)-plane and rotate it about the \(x\)-axis. Write the profile as a parametrized curve \((x(t),y(t))\), for \(a\le t\le b\). We will assume that \(y(t)\ge 0\), so the profile lies on or above the axis. The \(x\)-axis is the axis of revolution.

Definition: The profile curve is the plane curve being rotated. The axis of revolution is the fixed line around which it is rotated. The perpendicular radius at a profile point is the shortest distance from that point to the axis. For rotation about the \(x\)-axis, this radius is \(r(t)=y(t)\) when \(y(t)\ge 0\).

The radius is a distance, not a signed height. If a profile is allowed to lie on either side of the \(x\)-axis, its distance from the axis is \(|y(t)|\). Here we use a profile on or above the axis so that \(r(t)=y(t)\) throughout the setup.

We can describe the rotation using an angle \(\theta\), measured from the initial position in the \(xy\)-plane. A point on the rotating surface has coordinates

\mathbf{X}(t,\theta) = \bigl(x(t),\,y(t)\cos\theta,\,y(t)\sin\theta\bigr), \qquad 0\le \theta\le 2\pi.

When \(\theta=0\), the point is \((x(t),y(t),0)\), its original position in the profile plane. As \(\theta\) runs from \(0\) to \(2\pi\), the point traces a circle of radius \(y(t)\) around the \(x\)-axis. Thus the perpendicular radius of the circle is precisely the distance from the profile point to the axis.

The Local Arc-Length Element

A parameter change \(dt\) generally moves the profile point in both the \(x\)- and \(y\)-directions. Its local distance along the curve is therefore measured by arc length, not just by either coordinate change. For a differentiable profile, the speed is \(\sqrt{(x'(t))^2+(y'(t))^2}\), and the corresponding local arc-length element is

ds=\sqrt{(x'(t))^2+(y'(t))^2}\,dt.

This expression applies on a parameter interval where the curve is differentiable. If the profile is regular, meaning \((x'(t),y'(t))\ne(0,0)\), then its speed is positive. The formula also gives the local length element at a point where the speed is zero, but the parameterization may fail to describe a regular curve there.

The surface parameterization makes the relationship between the profile and the swept band precise. Differentiating with respect to \(t\) gives the direction along the profile at a fixed angle; differentiating with respect to \(\theta\) gives the direction around the circle:

\mathbf{X}_t = \bigl(x'(t),y'(t)\cos\theta,y'(t)\sin\theta\bigr), \qquad \mathbf{X}_\theta = \bigl(0,-y(t)\sin\theta,y(t)\cos\theta\bigr).

These directions are perpendicular. Indeed, their dot product is

\mathbf{X}_t\cdot\mathbf{X}_\theta = -y'(t)y(t)\cos\theta\sin\theta +y'(t)y(t)\sin\theta\cos\theta =0.

At the initial position \(\theta=0\), the angular tangent is \((0,0,y(t))\), which is perpendicular to the profile plane. At other angles, the angular tangent need not be perpendicular to that plane. It remains perpendicular to the profile tangent, as the dot-product calculation shows.

How a Short Segment Sweeps Out a Band

Imagine a very short portion of the profile, with arc length \(ds\). As it rotates, it forms a narrow band around the axis. At a point with radius \(r\), the circular direction has local length \(r\,d\theta\) for a small angular change \(d\theta\). The profile direction has local length \(ds\). Because the two directions are perpendicular, their local area factor is the product of these lengths.

This product can also be checked directly using the surface parameterization. The cross product of its two tangent vectors has magnitude

\left\lVert\mathbf{X}_t\times\mathbf{X}_\theta\right\rVert = y(t)\sqrt{(x'(t))^2+(y'(t))^2}.

To see this, the components of the cross product are \(\bigl(y(t)y'(t),-x'(t)y(t)\cos\theta,-x'(t)y(t)\sin\theta\bigr)\). Squaring and adding gives \(y(t)^2\bigl((y'(t))^2+(x'(t))^2\bigr)\), since \(\cos^2\theta+\sin^2\theta=1\). Taking the nonnegative square root gives the stated magnitude.

Thus the local area of the parameterized patch is \(r\,ds\,d\theta\). When the patch goes all the way around the axis, the angular direction covers a full circle. The circle has circumference \(2\pi r\), so the narrow band’s local area is represented by circumference times profile arc length. This is a local geometric description; it does not by itself compute the area of an entire surface. The area of a whole surface requires accounting for all the profile segments in the chosen interval.

Geometric interpretation: A short profile segment of length \(ds\), at perpendicular radius \(r\), sweeps out a narrow band. Its local area is determined by the circle of circumference \(2\pi r\) and the segment’s arc-length width \(ds\).

Worked Examples

Worked Example: A Slanted Profile Segment

Let the profile be \(x(t)=2t\), \(y(t)=1+t\), for \(0\le t\le1\), and rotate it about the \(x\)-axis. Find the radius and local arc-length element at \(t=\frac12\), then describe the band swept out by a very short parameter interval around that point.

The perpendicular radius is the profile’s height above the axis:

r\left(\frac12\right) =y\left(\frac12\right) =1+\frac12 =\frac32.

The derivatives are \(x'(t)=2\) and \(y'(t)=1\). Therefore the speed is

\sqrt{(x'(t))^2+(y'(t))^2} =\sqrt{2^2+1^2} =\sqrt5, \qquad ds=\sqrt5\,dt.

For example, a small interval of width \(\Delta t=0.01\) near \(t=\frac12\) has profile length approximately \(\sqrt5(0.01)=0.01\sqrt5\). The circle traced by the central point has circumference \(2\pi(\frac32)=3\pi\). The band is therefore locally approximated by a rectangle with area \(3\pi(0.01\sqrt5)=0.03\pi\sqrt5\).

As a check on the geometry, at \(\theta=\frac{\pi}{2}\) the two tangent vectors at \(t=\frac12\) are \(\mathbf{X}_t=(2,0,1)\) and \(\mathbf{X}_\theta=(0,-\frac32,0)\). Their dot product is \(2(0)+0(-\frac32)+1(0)=0\), confirming that the angular and profile directions are perpendicular. The radius, speed, and band-width calculation all use the stated parameterization.

Worked Example: A Curved Graph Profile

Take the graph \(y=1+x^2\), for \(0\le x\le1\), as a profile rotated about the \(x\)-axis. Find its radius and arc-length element at \(x=\frac13\), and estimate the local area of the band corresponding to a small horizontal interval \(\Delta x=0.03\).

Here the parameter is \(x\) itself, so \(x'=1\), \(y=1+x^2\), and \(y'=2x\). At \(x=\frac13\),

r=1+\left(\frac13\right)^2=\frac{10}{9}, \qquad y'=\frac23, \qquad ds=\sqrt{1+\left(\frac23\right)^2}\,dx =\frac{\sqrt{13}}{3}\,dx.

The profile length across the short interval is approximately \(\frac{\sqrt{13}}{3}(0.03)=\frac{\sqrt{13}}{100}\). The circle at the midpoint has circumference \(2\pi(\frac{10}{9})=\frac{20\pi}{9}\). The resulting local band-area estimate is

\frac{20\pi}{9}\cdot\frac{\sqrt{13}}{100} =\frac{\pi\sqrt{13}}{45}.

For a check, substitute \(x=\frac13\) into the graph and derivative: \(1+x^2=1+\frac19=\frac{10}{9}\) and \(2x=\frac23\). Substituting these values into the arc-length formula gives \(\sqrt{1+\frac49}=\sqrt{\frac{13}{9}}=\frac{\sqrt{13}}{3}\), as used above. Since the interval is short, the estimate uses the midpoint radius and local speed rather than treating the curved band as an exact rectangle.

Worked Example: A Straight Profile with a Nonunit Parameter Speed

Let \(x(t)=3t\) and \(y(t)=4t\), for \(1\le t\le2\), and rotate the profile about the \(x\)-axis. Find the radius and local arc-length element at \(t=1\). Then estimate the band area for a short interval \(\Delta t=0.02\) starting there.

The derivatives are \(x'=3\) and \(y'=4\), so the profile speed is constant:

\sqrt{(x')^2+(y')^2}=\sqrt{3^2+4^2}=5, \qquad ds=5\,dt.

At \(t=1\), the perpendicular radius is \(r=y(1)=4\). Across the short parameter interval, the profile length is exactly \(5(0.02)=0.1\), because the speed is constant. The circle at the starting point has circumference \(2\pi(4)=8\pi\). Using the starting radius for a local band estimate gives \(8\pi(0.1)=0.8\pi\).

To check the arc length directly, the profile points at \(t=1\) and \(t=1.02\) differ by \(\Delta x=3(0.02)=0.06\) and \(\Delta y=4(0.02)=0.08\). Their straight-line separation is \(\sqrt{0.06^2+0.08^2}=\sqrt{0.0036+0.0064}=\sqrt{0.01}=0.1\), agreeing with \(ds=5\,dt\) over this straight segment. The band estimate uses the radius at its starting point; the radius changes over the interval, so this estimate is local rather than an exact area for the full band.

A Common Pitfall: Using a Coordinate Change as the Band Width

A frequent error is to use only the vertical change \(dy\), or only the horizontal change \(dx\), as the width of a swept band. Neither is generally the length along a slanted profile. For the profile \(x(t)=2t\), \(y(t)=1+t\), we have \(dx=2\,dt\) and \(dy=dt\), but the profile length is

ds=\sqrt{(2\,dt)^2+(dt)^2}=\sqrt5\,dt.

At \(t=\frac12\), the radius is \(\frac32\). Using \(dy=dt\) as the width would incorrectly give a local band-area factor \(2\pi(\frac32)\,dt=3\pi\,dt\). The correct factor uses the arc-length element and is \(2\pi(\frac32)\sqrt5\,dt=3\pi\sqrt5\,dt\). The erroneous calculation omits the horizontal movement of the profile. The correction is to compute the local arc length from both coordinate changes before interpreting the width of the swept band.

Why the Geometry Matters

The same geometric quantities underlie many surface-of-revolution problems: the radius measures distance to the axis, and the local width comes from arc length along the profile. Keeping them distinct helps prevent a common setup error: replacing the perpendicular radius by a coordinate that is not a perpendicular distance, or replacing the profile length by a horizontal or vertical increment.

The parameterization also explains a useful feature of the geometry. At each point away from the axis, the direction along the profile and the direction around the circle are perpendicular. That is why the local patch has one length contribution from the profile and another from rotation. If the profile reaches the axis, the radius becomes zero and the angular direction collapses at that point; the simple nonzero-radius band picture then has to be interpreted with care.

Check Your Understanding

Use the definitions and local geometry above to answer these questions.

For a profile rotated about the \(x\)-axis, what is the perpendicular radius at a point \((x(t),y(t))\) with \(y(t)\ge0\)?
For \(x(t)=t\) and \(y(t)=2t+1\), what are the speed and local arc-length element?
Why is the angular tangent perpendicular to the profile tangent even when it lies in the profile plane?
For a short profile segment at radius \(r\), what two local lengths determine the narrow band’s area?
For a profile that crosses the axis, why is its perpendicular radius not always given by its signed \(y\)-coordinate?