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Parametrized Surfaces · Tutorial 836 of 999

Surfaces of Revolution From Generating Curves

Build surface parametrizations from generating curves and determine their parameter ranges, tangent directions, and boundary circles.

Intermediate 9 min read

What You'll Learn

  • Derive a surface parametrization by rotating a profile curve about each coordinate axis.
  • Choose an axial parameter and a nonnegative distance from the axis.
  • Find the two tangent directions from the partial derivatives.
  • Identify when those directions are independent and when the angular direction collapses.
  • Determine the circles or points traced by the profile endpoints.

From a Plane Curve to a Surface

A surface of revolution is formed by rotating a plane curve around a fixed axis. The curve is called the generating curve or profile. A point off the axis traces a circle as it rotates; a point on the axis stays fixed. The collection of these orbits sweeps out the surface.

The key to writing a parametrization is to describe each point of the profile by two pieces of information: its position along the axis and its distance from the axis. Rotation changes the direction around the axis while preserving both of those quantities.

Definition: A surface of revolution is the set obtained by rotating a plane curve about a fixed axis. A parametrization records a point on the profile and an angle of rotation, usually with the angle ranging from \(0\) to \(2\pi\).

Deriving the Coordinate Formulas

Begin with the \(z\)-axis. Describe a profile in a plane containing that axis by its axial coordinate \(q(s)\) and its distance \(\rho(s)\) from the axis, where \(\rho(s)\ge 0\). At angle \(0\), the point can be written as \((\rho(s),0,q(s))\). Rotation through angle \(\theta\) sends the first two coordinates around a circle while leaving the \(z\)-coordinate unchanged.

$$ \mathbf{S}(s,\theta) = \bigl(\rho(s)\cos\theta,\ \rho(s)\sin\theta,\ q(s)\bigr), \qquad 0\le \theta\le 2\pi. $$

The corresponding formulas for the other coordinate axes follow by placing the axial coordinate in the appropriate position. For the \(x\)-axis, the profile lies in a plane such as the \(xy\)-plane; for the \(y\)-axis, it can lie in the same plane. In each case, the remaining two coordinates describe the circular orbit.

$$ \begin{aligned} \text{About the }x\text{-axis:}\quad &\mathbf{S}(s,\theta) = \bigl(q(s),\ \rho(s)\cos\theta,\ \rho(s)\sin\theta\bigr),\\[4pt] \text{About the }y\text{-axis:}\quad &\mathbf{S}(s,\theta) = \bigl(\rho(s)\cos\theta,\ q(s),\ \rho(s)\sin\theta\bigr),\\[4pt] \text{About the }z\text{-axis:}\quad &\mathbf{S}(s,\theta) = \bigl(\rho(s)\cos\theta,\ \rho(s)\sin\theta,\ q(s)\bigr). \end{aligned} $$

Here \(s\) ranges over the profile, and \(\theta\) ranges over the angles used for the rotation. A full revolution uses \(0\le\theta\le2\pi\). Since the radius is a distance, use \(\rho(s)\ge0\). If a profile is initially described with a signed coordinate perpendicular to the axis, its distance from the axis is the absolute value of that coordinate.

Often the profile is a graph, so the axial coordinate itself can be used as the parameter. For example, a profile about the \(z\)-axis with radius \(r(z)\) gives \(\mathbf{S}(z,\theta)=(r(z)\cos\theta,r(z)\sin\theta,z)\). The more general \(q(s),\rho(s)\) form also allows profiles that are not graphs of radius as a function of axial coordinate.

Tangent Directions

For a parametrized surface, the partial derivatives with respect to its parameters give tangent vectors wherever the derivatives exist. For rotation about the \(z\)-axis, differentiating with respect to the profile parameter and the angle gives

$$ \mathbf{S}_s = \bigl(\rho'(s)\cos\theta,\ \rho'(s)\sin\theta,\ q'(s)\bigr), \qquad \mathbf{S}_\theta = \bigl(-\rho(s)\sin\theta,\ \rho(s)\cos\theta,\ 0\bigr). $$

The first vector follows the profile as \(s\) changes; the second follows the circular orbit as \(\theta\) changes. The same interpretation applies for rotation about the \(x\)- or \(y\)-axis, with the coordinates placed according to the corresponding parametrization.

When \(\rho(s)>0\), the angular tangent vector is nonzero. If the profile is regular, meaning \(q'(s)\) and \(\rho'(s)\) are not both zero, the two tangent vectors are independent. One way to verify this is to take their cross product: its magnitude is \(\rho(s)\sqrt{q'(s)^2+\rho'(s)^2}\), which is positive under those conditions. At \(\rho(s)=0\), the angular derivative is the zero vector because all angles describe the same point on the axis. This is a coordinate collapse, not a circular orbit of positive radius.

Worked Example: Rotating a Parabolic Profile About the \(x\)-Axis

Worked Example: Rotating a Parabolic Profile About the \(x\)-Axis

Take the profile \(x=t^2,\ y=t\), for \(0\le t\le2\), in the \(xy\)-plane, and rotate it about the \(x\)-axis. The axial coordinate is \(q(t)=t^2\), and the distance from the axis is \(\rho(t)=t\). Substituting these into the \(x\)-axis formula gives

$$ \mathbf{S}(t,\theta) = \bigl(t^2,\ t\cos\theta,\ t\sin\theta\bigr), \qquad 0\le t\le2,\quad 0\le\theta\le2\pi. $$

The tangent directions are

$$ \mathbf{S}_t=(2t,\cos\theta,\sin\theta), \qquad \mathbf{S}_\theta=(0,-t\sin\theta,t\cos\theta). $$

For example, at \(t=1,\theta=0\), these are \((2,1,0)\) and \((0,0,1)\). Substitution into the surface parametrization gives the point \((1,1,0)\), so both tangent directions are based at that point.

At \(t=0\), the profile meets the axis and every angle gives \((0,0,0)\), a single point. At \(t=2\), the endpoint traces the circle \((4,2\cos\theta,2\sin\theta)\), of radius \(2\) in the plane \(x=4\). As a check on the surface described, \(x=t^2\) and \(y^2+z^2=t^2\), so its points satisfy \(y^2+z^2=x\), with \(0\le x\le4\). Conversely, for any such \(x\), setting \(t=\sqrt{x}\) gives radius \(t\), so the parametrization reaches the corresponding circle.

Worked Example: A Curved Profile About the \(z\)-Axis

Worked Example: A Curved Profile About the \(z\)-Axis

Let the profile have axial coordinate \(q(t)=t\) and radius \(\rho(t)=2+t^2\), for \(-1\le t\le1\). Rotating about the \(z\)-axis gives

$$ \mathbf{S}(t,\theta) = \bigl((2+t^2)\cos\theta,\ (2+t^2)\sin\theta,\ t\bigr), \qquad -1\le t\le1,\quad 0\le\theta\le2\pi. $$

Differentiating each component yields

$$ \mathbf{S}_t=(2t\cos\theta,2t\sin\theta,1), \qquad \mathbf{S}_\theta=(-(2+t^2)\sin\theta,(2+t^2)\cos\theta,0). $$

At \(t=0,\theta=0\), the point is \((2,0,0)\); the tangent vectors are \((0,0,1)\) and \((0,2,0)\). The profile endpoints \(t=-1\) and \(t=1\) give circles of radius \(3\), at \(z=-1\) and \(z=1\), respectively.

To check the surface equation, note that \(z=t\) and \(x^2+y^2=(2+t^2)^2(\cos^2\theta+\sin^2\theta)=(2+t^2)^2\). Thus \(x^2+y^2=(2+z^2)^2\) for \(-1\le z\le1\), as expected from the profile. The trigonometric identity also confirms that the distance from the \(z\)-axis is \(2+t^2\) at every angle.

Worked Example: A Profile About the \(y\)-Axis

Worked Example: A Profile About the \(y\)-Axis

Consider the profile \(x=4-y^2\) for \(-1\le y\le1\), in the \(xy\)-plane. Its axial coordinate is \(q(u)=u\), and its radius from the \(y\)-axis is \(\rho(u)=4-u^2\). Therefore the parametrization is

$$ \mathbf{S}(u,\theta) = \bigl((4-u^2)\cos\theta,\ u,\ (4-u^2)\sin\theta\bigr), \qquad -1\le u\le1,\quad 0\le\theta\le2\pi. $$

The two tangent directions are

$$ \mathbf{S}_u=(-2u\cos\theta,1,-2u\sin\theta), \qquad \mathbf{S}_\theta=(-(4-u^2)\sin\theta,0,(4-u^2)\cos\theta). $$

At \(u=0,\theta=0\), the point is \((4,0,0)\), and the tangent vectors are \((0,1,0)\) and \((0,0,4)\). At either profile endpoint, \(u=-1\) or \(u=1\), the radius is \(4-1=3\). The endpoints therefore sweep out circles of radius \(3\) in the planes \(y=-1\) and \(y=1\).

For a consistency check, the parametrization gives \(y=u\) and \(x^2+z^2=(4-u^2)^2(\cos^2\theta+\sin^2\theta)=(4-u^2)^2\). Hence \(x^2+z^2=(4-y^2)^2\), with \(-1\le y\le1\), matching the profile's radius at each axial position.

Parameter Ranges and Boundary Circles

The parameter domain specifies which portion of the profile is rotated and how much of each orbit is included. If \(s\) ranges from \(a\) to \(b\) and \(\theta\) runs through a full revolution, the profile endpoints at \(s=a\) and \(s=b\) determine the end curves of the swept patch. For example, about the \(z\)-axis, fixing \(s=s_0\) gives

$$ \bigl(\rho(s_0)\cos\theta,\ \rho(s_0)\sin\theta,\ q(s_0)\bigr), \qquad 0\le\theta\le2\pi. $$

This is a circle centered on the axis, in the plane \(z=q(s_0)\), with radius \(\rho(s_0)\). If \(\rho(s_0)=0\), every value of \(\theta\) gives the same point, so that end of the patch is a point rather than a boundary circle of positive radius. The descriptions for the other axes are the corresponding coordinate permutations: the circle lies in a plane perpendicular to the axis of rotation.

For a full revolution, \(\theta=0\) and \(\theta=2\pi\) describe the same points. The two sides of the rectangular parameter domain at those angles are therefore a seam in the parametrization, not two distinct geometric boundary curves. If only a restricted range of angles is used, the corresponding angular edges can be boundaries of the swept patch.

Why the Rotation Formula Works

The formula can be justified directly from the geometry of rotation. For the \(z\)-axis, a profile point with axial coordinate \(q(s)\) and radius \(\rho(s)\) starts at \((\rho(s),0,q(s))\). Rotating its first two coordinates by angle \(\theta\) gives \((\rho(s)\cos\theta,\rho(s)\sin\theta)\); the third coordinate is unchanged. Thus each profile point maps to exactly its circular orbit, or to a fixed point if its radius is zero. Letting \(s\) run over the whole profile collects all those orbits and produces the surface.

Key takeaway: To parametrize a surface of revolution, preserve the profile's axial coordinate, use its nonnegative distance from the axis as the orbit radius, and let the angle trace the orbit. The profile endpoints then determine the end circles, unless an endpoint lies on the axis and collapses to a point.

A Common Coordinate Mistake

A frequent error is to put the axial coordinate in the wrong component. Suppose the profile is \(x=2,\ z=t\), with \(0\le t\le1\), and is rotated about the \(z\)-axis. The axial coordinate is \(t\), and the radius is \(2\), so the correct parametrization is

$$ \mathbf{S}(t,\theta)=(2\cos\theta,2\sin\theta,t). $$

A wrong calculation might instead write \((t,2\cos\theta,2\sin\theta)\). That formula puts \(t\) along the \(x\)-axis and rotates around the \(x\)-axis; it describes a different surface. Check the intended axis by asking which coordinate must stay fixed during each circular orbit. Here \(z=t\) must remain fixed while \(x^2+y^2=4\), which confirms the correct formula.

Why These Parametrizations Matter

Parametrizing a surface from its generating curve turns a geometric description into data that can be used in later calculations. The parameters identify the profile and the circular direction; their partial derivatives identify the two natural tangent directions; and the parameter limits show where the swept patch begins and ends. Keeping the axial coordinate, radial distance, and angle distinct makes the geometry easier to check and prevents axis and boundary errors.

Check Your Understanding

Use the formulas and geometric checks from this tutorial to answer the following.

  1. A profile about the \(z\)-axis has axial coordinate \(q(s)=s\) and radius \(\rho(s)=3-s\), for \(0\le s\le2\). Write its parametrization and state the radii and heights of its endpoint circles.
  2. For rotation about the \(x\)-axis, which coordinate stays fixed as the angle changes?
  3. In the \(z\)-axis formula, what does \(\mathbf{S}_\theta\) represent geometrically?
  4. What orbit is traced by a profile point whose distance from the axis is zero?
  5. Why do the parameter edges \(\theta=0\) and \(\theta=2\pi\) describe a seam, rather than distinct curves, in a full revolution?

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