Ellipses by Scaling Circular Parametrizations

From a Circle to an Ellipse

A circle can be parametrized using cosine and sine because \(\cos^2 t+\sin^2 t=1\). An ellipse with different horizontal and vertical semiaxis lengths has the same basic structure, but the two circular coordinates must be scaled separately. This gives a direct way to write parametrizations for ellipses in the coordinate planes and to control which part of an ellipse is traced.

The center of an ellipse determines where the curve is located. Its two positive semiaxis lengths determine how far it extends in each coordinate direction. The parameter interval then determines which points are traced and in what order.

Definition and General Forms

An axis-aligned ellipse in a coordinate plane is the curve whose equation, after shifting to its center, has the form \(u^2/a^2+v^2/b^2=1\), where \(a>0\) and \(b>0\). The positive numbers \(a\) and \(b\) are the semiaxis lengths in the respective coordinate directions. They need not be equal; if they are equal, the ellipse is a circle.

For a parametrization, the point on the curve is given by a vector-valued function \(\mathbf{r}(t)\). The following forms use \(t\) as the angle parameter. The center coordinates are added to the scaled cosine and sine components.

In each line, \(h\) and \(k\) are the center coordinates within the named plane. The first semiaxis \(a\) is along the first varying coordinate in that plane, and \(b\) is along the second varying coordinate. For example, in the \(xz\) plane, \(a\) measures the extent in the \(x\)-direction and \(b\) measures the extent in the \(z\)-direction. The fixed coordinate \(c\) determines which parallel coordinate plane contains the ellipse.

For a complete traversal, use \(0\le t\le 2\pi\). The points at \(t=0\) and \(t=2\pi\) coincide, so the endpoint is repeated. Other intervals can trace arcs; their choice also affects the direction of travel.

Why Separate Scaling Produces an Ellipse

The standard unit-circle parametrization is \(\langle \cos t,\sin t\rangle\). Its coordinates satisfy \(u^2+v^2=1\). Multiplying the first coordinate by \(a\) and the second by \(b\) changes this equation into the ellipse equation. Adding the center shifts the curve without changing its shape.

This argument shows both that every point produced by the parametrization lies on the ellipse and that the parametrization reaches every point of the ellipse over a complete cycle. It also explains why the scale factors belong to their respective coordinates: each normalized coordinate must divide by its own semiaxis length.

Worked Example: A Complete Ellipse in the \(xy\) Plane

Choosing an Interval to Select an Arc

The parameter interval determines which angles are used. Because cosine and sine change sign in different quadrants, intervals such as \([0,\pi]\) and \([\pi,2\pi]\) select different halves for the standard ordering \(x=h+a\cos t,\ y=k+b\sin t\). On \([0,\pi]\), \(\sin t\ge 0\), so the parametrization stays on or above the center’s horizontal line. On \([\pi,2\pi]\), \(\sin t\le 0\), so it stays on or below that line.

The endpoints and direction should be checked rather than inferred from the interval name alone. Evaluate \(\mathbf{r}\) at both endpoints. To check which side of the ellipse is traced, inspect the signs of the relevant sine or cosine term throughout the interval. To check direction, compare the endpoint coordinates or use the derivatives of the component formulas.

Worked Example: The Lower Half from Largest \(y\) to Smallest \(y\)

Worked Example: A Quarter-Ellipse in the \(xz\) Plane

A Common Scaling Error

A frequent mistake is to put a semiaxis length in the denominator of the wrong squared coordinate. For example, the parametrization \(x=2\cos t,\ y=5\sin t\) has semiaxis \(2\) in the \(x\)-direction and semiaxis \(5\) in the \(y\)-direction. Its equation is \(x^2/4+y^2/25=1\), not \(x^2/25+y^2/4=1\).

The incorrect equation fails even at the parameter value \(t=0\): the parametrization gives \((x,y)=(2,0)\), and substitution into the incorrect equation gives \(2^2/25+0^2/4=4/25\ne1\). In the correct equation, the same point gives \(2^2/4+0^2/25=1\). The denominator for each coordinate must be the square of that coordinate’s semiaxis length.

Another interval error is to use \([0,\pi]\) automatically for a lower half. With \(y=k+b\sin t\), that interval gives \(y\ge k\), which is the upper half. The sign in the component formula matters: in the second worked example, the lower half is selected because \(z=-1-5\sin t\), not \(-1+5\sin t\). Checking the component formula and endpoints prevents both interval and direction errors.

Why This Construction Is Useful

Separate scaling makes an ellipse parametrization nearly as direct as a circle parametrization. It identifies the center, the two semiaxis lengths, the containing coordinate plane, and the traversal order in a single vector-valued function. The same verification works in each coordinate plane: subtract the center coordinates, divide by the corresponding semiaxis lengths, and use \(\cos^2 t+\sin^2 t=1\).

For a complete ellipse, use an interval of length \(2\pi\), such as \([0,2\pi]\). For a half or quarter, choose endpoints that delimit the desired angle range, then verify the resulting points and signs. When direction is specified, also check the endpoint order; an interval that traces the right arc can still travel in the wrong direction.