Introduction
A plane curve can be described by giving a point’s horizontal and vertical coordinates as functions of a parameter. As the parameter varies over a specified interval, those coordinates determine a collection of points in the plane.
The formulas are one way of describing the curve, but they are not the curve’s geometric trace itself. Different pairs of formulas can produce the same set of points. Keeping these ideas distinct is essential: a parametrization specifies how points are generated, while the trace is the set of points generated.
Definition of a Parametrized Plane Curve
Let \(I\) be an interval of real numbers, and let \(x:I\to\mathbb{R}\) and \(y:I\to\mathbb{R}\) be functions. For each \(t\in I\), the formulas assign the point \((x(t),y(t))\) in the plane.
The parameter \(t\) is an input to both coordinate functions. The interval \(I\) is part of the description: only values of \(t\) in that interval are allowed. The ordered pair \((x(t),y(t))\) gives the point corresponding to a particular allowed parameter value.
It is useful to distinguish three related things:
- The parameter interval: the allowed values of \(t\).
- The parametrization: the coordinate rules \(x(t)\) and \(y(t)\), used on that interval.
- The trace: the set of plane points obtained from those rules.
In particular, a parametrization is not just an equation involving \(x\) and \(y\). It includes the parameter interval. And the trace is a set: if more than one parameter value produces the same point, that point still appears only once in the set.
Finding Points from a Parametrization
To find the point associated with a particular parameter value, substitute that value into both coordinate functions. For example, if \(x(t)=1+2t\) and \(y(t)=3-t\), then at \(t=1\) the point is \((1+2,3-1)=(3,2)\). The same parameter value must be used in both coordinates.
Worked Example: Tracing a Line Segment
Consider \(x(t)=1+2t\), \(y(t)=3-t\), with \(t\in[0,2]\). Find the points corresponding to the endpoints of the interval and to \(t=1\), and describe the trace.
At the left endpoint \(t=0\), substitute into each coordinate:
The corresponding point is \((1,3)\). At \(t=1\),
so the point is \((3,2)\). At the right endpoint \(t=2\),
The endpoint is \((5,1)\). To see that the trace includes exactly the straight segment between the endpoints, write \(t=2\lambda\), where \(\lambda\in[0,1]\). Then
As \(\lambda\) ranges from \(0\) to \(1\), this gives every point on the segment from \((1,3)\) to \((5,1)\), and no point outside it. Checking \(\lambda=0\) gives \((1,3)\); checking \(\lambda=1\) gives \((5,1)\); and \(\lambda=\tfrac12\) gives \((3,2)\), agreeing with the direct substitution at \(t=1\).
The Interval Limits Which Points Are Included
The same coordinate formulas can produce different traces when used on different intervals. A formula by itself does not say which parameter values are allowed, so it cannot always determine the intended curve. The interval also tells us whether its endpoints are included: for example, \(t\in[0,1]\) allows \(t=0\) and \(t=1\), while \(t\in(0,1)\) does not.
Worked Example: A Curved Arc on a Restricted Interval
Let \(x(t)=t^2\) and \(y(t)=t\), with \(t\in[-1,1]\). Find the points at \(t=-1\), \(t=0\), and \(t=1\), and explain what the interval contributes to the description.
At \(t=-1\),
so the point is \((1,-1)\). At \(t=0\),
giving \((0,0)\). At \(t=1\),
giving \((1,1)\). These substitutions check each coordinate directly. Since the parameter is restricted to \([-1,1]\), the trace is the curved portion generated between these parameter values, including the two endpoint points. If the interval were different, the allowed inputs—and therefore potentially the set of generated points—would need to be reconsidered.
One Trace Can Have Different Parametrizations
The trace is determined by the set of output points, not by the names or formulas used for the parameter. Thus, two descriptions may use different parameter values and different coordinate rules while still generating exactly the same set. To establish that they do, one must check that each description produces all and only the points in that set.
Worked Example: Two Descriptions of the Same Segment
Compare these two parametrizations:
- \(x(t)=t,\ y(t)=2t+1,\ t\in[0,1]\);
- \(x(u)=u^2,\ y(u)=2u^2+1,\ u\in[0,1]\).
For the first description, the endpoint values are
For the second description, the endpoint values are
Matching endpoints alone would not prove that the full traces agree. For the first parametrization, as \(t\) ranges over \([0,1]\), its first coordinate \(t\) takes every value in \([0,1]\). For the second, \(u^2\) also takes every value in \([0,1]\): it never leaves that interval, and for any \(t\in[0,1]\), choosing \(u=\sqrt{t}\) gives \(u\in[0,1]\) and \(u^2=t\).
Both descriptions therefore produce precisely the points \((s,2s+1)\) with \(s\in[0,1]\). For instance, the point \((\tfrac14,\tfrac32)\) comes from \(t=\tfrac14\) in the first description, since \(2(\tfrac14)+1=\tfrac32\). In the second description, it comes from \(u=\tfrac12\), since \((\tfrac12)^2=\tfrac14\) and \(2(\tfrac14)+1=\tfrac32\). These substitutions check the shared point, while the range argument proves equality of the entire traces.
Why the Interval Must Not Be Ignored
A common error is to use a relation suggested by the coordinate formulas while ignoring the specified interval. This can lead to claiming that a parametrization produces more points than it actually does.
Common Mistake: Forgetting the Allowed Parameter Values
Take \(x(t)=t\), \(y(t)=t^2\), with \(t\in[0,1]\). A mistaken calculation might say: “Since \(y=t^2\) and \(x=t\), the curve is the entire parabola \(y=x^2\).” The substitution \(t=x\) does give \(y=x^2\), but it also requires \(x=t\in[0,1]\). The conclusion that the entire parabola is traced omits that restriction.
For example, the point \((-1,1)\) satisfies \(y=x^2\). But to produce it from the parametrization, its first coordinate would require \(t=-1\), which is not in \([0,1]\). Thus this point is not on the trace. The correction is to retain the parameter interval when describing the generated points: only the portion produced for \(0\le t\le1\) is included. Direct endpoint checks confirm that \(t=0\) gives \((0,0)\) and \(t=1\) gives \((1,1)\).
What the Parametric Description Tells You
A parametric description gives a systematic way to generate points: choose an allowed parameter value, then evaluate both coordinate functions. It also makes clear which parameter values are included. These are the basic ingredients needed before studying further properties of plane curves.
When reading or writing a parametrization, check that the interval is stated, that both coordinates use the same parameter, and that any description of the trace respects the allowed values. When comparing two parametrizations, matching a few sample points is not enough to establish that their traces agree; the comparison must account for all points each one produces.
Check Your Understanding
Use the definitions and examples above to answer the following questions.
- For \(x(t)=2-t\), \(y(t)=t+1\), with \(t\in[0,2]\), what point corresponds to \(t=1\)?
- In the definition of a parametrized plane curve, what role does the interval \(I\) play?
- Can two different pairs of coordinate functions produce the same geometric trace? Explain what must be checked to establish that they do.
- For \(x(t)=t\), \(y(t)=t^2\), with \(t\in[0,1]\), why is the point \((-1,1)\) not on the trace?
- Why does substituting a few parameter values not by itself prove that two parametrizations have the same trace?
