Restricting Parametrizations to Arcs

Introduction

A parametrization describes a curve as a parameter varies. Restricting the parameter to an interval selects a particular portion of that curve. This is how we specify an arc rather than the entire circle, ellipse, or helix.

The endpoints alone do not always determine the requested arc: two points on a circle, for example, lie on two arcs. The parameter limits also determine which portion is traced and, as the parameter increases, the direction of traversal.

Restricting a Parametrization

Let \(\mathbf r(t)\) be a parametrization defined for all parameter values in an interval containing \([a,b]\), where \(a\le b\). Restricting it to \([a,b]\) means using the same formula but allowing only \(a\le t\le b\). The selected curve is the set of points \(\mathbf r(t)\) for \(t\in[a,b]\). The initial point is \(\mathbf r(a)\), and the terminal point is \(\mathbf r(b)\).

The interval is closed when the intended arc includes both endpoints. As \(t\) increases from \(a\) to \(b\), the parametrization traces the arc in its specified direction. Changing the interval can select a different arc, even when its endpoints are unchanged.

Finding Angle Limits on a Circle

For a circle of radius \(R\) centered at \((h,k)\) in the \(xy\)-plane, a standard parametrization is \(\mathbf r(t)=(h+R\cos t,\ k+R\sin t,\ 0)\). The parameter \(t\) is an angle measured from the positive \(x\)-direction relative to the circle’s center. Increasing \(t\) moves counterclockwise in the plane.

If an endpoint is \((x,y,0)\), its angle satisfies \(\cos t=(x-h)/R\) and \(\sin t=(y-k)/R\). Both coordinates matter: knowing only a cosine does not distinguish angles on opposite sides of the circle. Once the endpoint angles are found, choose values of \(t\) in the requested direction. If necessary, add or subtract a full turn, \(2\pi\), to represent the intended arc.

Endpoint Angles and Ambiguous Arcs

On a circle, two distinct endpoints usually determine two possible arcs: one shorter and one longer. The endpoint coordinates give angles modulo \(2\pi\), but the desired route determines which representatives to use. For counterclockwise traversal, choose an increasing angle interval. If the terminal angle appears smaller than the initial angle, adding \(2\pi\) gives an equivalent angle after one full turn. For clockwise traversal with the standard circle parametrization, the angle must decrease; one convenient way to keep a parameter interval increasing is to replace \(t\) by \(-s\).

For example, the parametrization \((\cos t,\sin t,0)\) has the point \((1,0,0)\) at \(t=0\), and the point \((0,1,0)\) at \(t=\pi/2\). The interval \([0,\pi/2]\) gives the short counterclockwise arc. It does not give the long clockwise arc between those same points.

Restricting an Ellipse

An ellipse obtained by scaling a circle can be parametrized as \(\mathbf r(t)=(h+a\cos t,\ k+b\sin t,\ z_0)\), where \(a>0\) and \(b>0\) are its horizontal and vertical semiaxis lengths. The parameter still records the angle in the underlying circle, not generally the geometric angle of the point as viewed from the ellipse’s center. To select an arc, use the endpoint coordinates to solve for \(\cos t\) and \(\sin t\), then choose the interval that traces the desired portion.

Using Height to Restrict a Helix

A helix with constant radius and a height that increases linearly with its parameter can be written in the form \(\mathbf r(t)=(R\cos t,\ R\sin t,\ z_0+ct)\), where \(R>0\) and \(c\ne0\). The first two coordinates determine the position around the circular axis; the third determines the height. Because the height is linear in \(t\), a specified height range can be converted directly into parameter limits.

Why the Restriction Selects the Arc

The reason interval selection works is that the parametrization already describes the full curve as its parameter varies. Keeping only the parameter values in \([a,b]\) keeps only the corresponding points. For the standard circle, \((\cos t,\sin t)\) runs around the circle as \(t\) increases, and its coordinates repeat after \(2\pi\). Therefore, an interval of length less than \(2\pi\) selects the portion swept out during that angular change; an interval of length \(2\pi\) makes a full revolution.

The same principle applies to a scaled circle and to a helix: changing the allowed parameter values changes which points of the already-defined curve are included. For an ellipse, a full \(2\pi\) parameter range makes one complete circuit. For a helix, the same parameter change makes one full revolution while also changing the height by the corresponding amount.

A Common Mistake: Confusing Endpoints with the Arc

Consider the unit circle \(\mathbf r(t)=(\cos t,\sin t,0)\), and suppose the requested path starts at \((1,0,0)\), ends at \((0,1,0)\), and goes clockwise along the longer arc. A tempting choice is \(t\in[0,\pi/2]\), since its endpoint values are correct. But this interval traces the short counterclockwise quarter-circle, not the requested route.

To make the parameter increase while the angle decreases, set \(t=-s\), where \(s\in[0,3\pi/2]\). The resulting parametrization is \(\mathbf q(s)=(\cos s,-\sin s,0)\). At the start, \(\mathbf q(0)=(1,0,0)\). At the end, \(\mathbf q(3\pi/2)=(0,1,0)\). As \(s\) increases through \(3\pi/2\), the angle in the original circle parametrization decreases by \(3\pi/2\), tracing the longer clockwise arc.