From a Curve to a Journey
A parametrization gives more information than the shape of a curve. It also tells us how a moving point travels along that curve as the parameter increases. The same geometric curve can be traced in different directions, or more than once, depending on the formulas and the parameter interval.
To read that information, keep track of the point’s position in parameter order. A list of the positions at selected increasing values of \(t\), with arrows between them, often makes the orientation clear. Comparing positions at different parameter values can also reveal repeated tracing.
Definitions: Orientation and Repeated Tracing
Let \(x(t)\) and \(y(t)\) be coordinate functions defined on a parameter interval \(I\). The parametrized point is \(\mathbf r(t)=(x(t),y(t))\). As \(t\) increases through \(I\), the point moves along the curve.
The trace on \(I\) is the set of points \(\{\mathbf r(t):t\in I\}\). The trace alone does not record the order in which the point visits those points. To determine orientation or repeated tracing, we must consider both the trace and the parameter values.
A curve can also pass through one point at more than one parameter value without tracing the entire curve more than once. Such a point is a self-intersection if distinct parts of the path meet there. The distinction matters: repeated tracing concerns revisiting a path, while a self-intersection concerns paths meeting at a point.
Read Orientation from Ordered Values
Begin by choosing parameter values in increasing order. Evaluate both coordinate functions at each value, then connect the resulting points in that order. These points need not describe every detail of the curve, but well-chosen values can show its direction clearly.
Orientation is relative to the direction in which \(t\) increases on the interval being considered.
Make a short table of \(t\), \(x(t)\), \(y(t)\), and \(\mathbf r(t)\).
Use arrows to indicate which point comes next as \(t\) increases.
Look for different parameter values giving the same point or for a repeated pattern in both coordinate functions.
Worked Example: Orientation Around a Circle
Worked Example: Orientation Around a Circle
Consider \(x(t)=2\cos t\), \(y(t)=2\sin t\) for \(0\le t\le 2\pi\). Determine the direction of travel using key parameter values.
At \(t=0\), \(\cos 0=1\) and \(\sin 0=0\), so \(\mathbf r(0)=(2,0)\).
At \(t=\frac{\pi}{2}\), \(\cos\frac{\pi}{2}=0\) and \(\sin\frac{\pi}{2}=1\), so \(\mathbf r\left(\frac{\pi}{2}\right)=(0,2)\).
At \(t=\pi\), \(\cos\pi=-1\) and \(\sin\pi=0\), so \(\mathbf r(\pi)=(-2,0)\).
At \(t=\frac{3\pi}{2}\), \(\cos\frac{3\pi}{2}=0\) and \(\sin\frac{3\pi}{2}=-1\), so \(\mathbf r\left(\frac{3\pi}{2}\right)=(0,-2)\).
Finally, at \(t=2\pi\), \(\cos 2\pi=1\) and \(\sin 2\pi=0\), so \(\mathbf r(2\pi)=(2,0)\).
The point moves from the right side of the circle to the top, then to the left, then to the bottom, and back to the right. This is counterclockwise travel. The endpoint agrees with the starting point, and the ordered intermediate values show one complete traversal, not two. As a check, the coordinates satisfy \(x(t)^2+y(t)^2=4\) at every parameter value, since \(4\cos^2 t+4\sin^2 t=4\); thus the listed positions all lie on the same circle.
Periodicity Explains Repeated Traversals
A reliable way to detect a repeated path is to check whether both coordinate functions repeat after the same positive parameter shift. If they do, the point returns to its earlier position and then repeats the same motion.
Here is the justification. For every eligible \(t\), the two coordinate equalities give
Thus each parameter value in one block corresponds to a parameter value \(P\) units later with exactly the same position. As \(t\) increases through the later block, the corresponding earlier values also increase in order, so the path is repeated in the same direction. On an interval of length \(2P\), the two blocks therefore give two traversals, with the shared endpoint counted as the meeting point between them.
Worked Example: A Circle Traced Twice
Worked Example: A Circle Traced Twice
Let \(x(t)=\cos(2t)\), \(y(t)=\sin(2t)\) for \(0\le t\le2\pi\). Determine how many times the circle is traced.
The coordinate functions repeat after a shift of \(\pi\), because
So the period of this parametrization is \(\pi\). The interval from \(0\) to \(2\pi\) contains two such periods. The key positions make the repetitions visible:
At \(t=0\), \(\mathbf r(0)=(1,0)\). At \(t=\frac{\pi}{4}\), \(\mathbf r\left(\frac{\pi}{4}\right)=(0,1)\). At \(t=\frac{\pi}{2}\), \(\mathbf r\left(\frac{\pi}{2}\right)=(-1,0)\). At \(t=\frac{3\pi}{4}\), \(\mathbf r\left(\frac{3\pi}{4}\right)=(0,-1)\). At \(t=\pi\), \(\mathbf r(\pi)=(1,0)\) again.
These values show one trip around the circle from \((1,0)\) back to \((1,0)\). Since \(\mathbf r(t+\pi)=\mathbf r(t)\), the next interval from \(\pi\) to \(2\pi\) repeats that same ordered trip. As a check, the point remains on the unit circle because \(\cos^2(2t)+\sin^2(2t)=1\), and \(\mathbf r(2\pi)=(1,0)\) is the starting position. The interval therefore traces the circle twice, both times counterclockwise.
Worked Example: A Segment Traced Out and Back
Worked Example: A Segment Traced Out and Back
Consider \(x(t)=\cos t\), \(y(t)=0\) for \(0\le t\le2\pi\). Describe the motion and identify the repeated portion.
At \(t=0\), \(\mathbf r(0)=(1,0)\). At \(t=\frac{\pi}{2}\), \(\mathbf r\left(\frac{\pi}{2}\right)=(0,0)\). At \(t=\pi\), \(\mathbf r(\pi)=(-1,0)\). As \(t\) goes from \(0\) to \(\pi\), the cosine coordinate decreases from \(1\) to \(-1\), so the point travels along the segment from right to left.
For the second half, at \(t=\frac{3\pi}{2}\), \(\mathbf r\left(\frac{3\pi}{2}\right)=(0,0)\), and at \(t=2\pi\), \(\mathbf r(2\pi)=(1,0)\). The cosine coordinate increases from \(-1\) to \(1\) as \(t\) goes from \(\pi\) to \(2\pi\), so the point travels back along the same segment, from left to right.
Every interior point of the segment is visited once on the way left and again on the way right. The endpoints are the turning positions: \((-1,0)\) at \(t=\pi\) and \((1,0)\) at \(t=0,2\pi\). As a check, every listed point has \(y=0\) and \(-1\le x\le1\), so all lie on the same segment. This is repeated tracing in opposite directions, rather than two traversals in the same direction.
A Self-Intersection Is Not Necessarily a Repeated Curve
Two different parameter values can produce the same point even when the path between those values is different. For instance, take \(x(t)=\sin t\), \(y(t)=\sin(2t)\) on \(0\le t\le2\pi\). At \(t=0,\pi,2\pi\), the point is \((0,0)\). At \(t=\frac{\pi}{2}\), it is \((1,0)\), and at \(t=\frac{3\pi}{2}\), it is \((-1,0)\).
From \(0\) to \(\pi\), \(\sin t\ge0\), so that part of the path lies on the side where \(x\ge0\). From \(\pi\) to \(2\pi\), \(\sin t\le0\), so the other part lies where \(x\le0\). The two portions meet at the origin, but they lie on opposite sides apart from that shared point. The origin is revisited; the whole first portion is not traced again. This is why a repeated point alone does not establish repeated tracing.
Why Orientation and Repetition Matter
Orientation is part of the information encoded by a parametrization. Two parametrizations may have the same trace but opposite directions, or one may cover that trace more times than the other. These distinctions matter whenever a problem asks how a point moves, rather than merely which points belong to a curve.
For AP problems, a useful habit is to state the parameter interval explicitly and support conclusions with ordered coordinate values. When claiming repeated full tracing, establish that both coordinates repeat together and explain how many repeats fit in the interval. When identifying a self-intersection, name the distinct parameter values that produce the shared point, and avoid concluding that the whole path repeats unless the intervening motion confirms it.
Check Your Understanding
Use ordered parameter values and coordinate comparisons to answer each question.
- For \(x(t)=\sin t\), \(y(t)=\cos t\) on \(0\le t\le2\pi\), find the positions at \(t=0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi\), and state the orientation.
- If both coordinate functions satisfy \(x(t+4)=x(t)\) and \(y(t+4)=y(t)\), how many repeated blocks of motion occur on an interval of length \(12\)?
- For \(x(t)=\cos t\), \(y(t)=0\) on \(0\le t\le2\pi\), which point is reached halfway through the interval, and what does that indicate about the direction on the second half?
- Explain why \(\mathbf r(a)=\mathbf r(b)\) for \(a\ne b\) does not by itself prove that the curve was traced twice.
- For \(x(t)=\cos(3t)\), \(y(t)=\sin(3t)\) on \(0\le t\le2\pi\), find a positive period of the coordinate pair and determine how many periods fit in the interval.
