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Reading a Parametrization Through Sample Values

Evaluate a space-curve parametrization at chosen parameter values and read the resulting ordered points carefully.

Beginner 9 min read

What You'll Learn

  • Substitute a parameter value into every component of a vector-valued function.
  • Record each result as an ordered point in three-dimensional space.
  • Use a sequence of sample values to describe sampled locations and progression.
  • Distinguish the parameter from the coordinates of the point it produces.
  • Recognize what a finite set of samples can and cannot tell you about a curve.

From a Formula to Points in Space

A vector-valued function gives a point in space for each allowed parameter value. Its formula may be compact, but evaluating it at specific values turns that formula into concrete locations that can be read, compared, and recorded in order.

For now, the aim is to make and interpret these evaluations. A few sample points can show where a parametrization places the curve at chosen parameter values and how those sampled locations change as the values increase. They do not, by themselves, describe every point between the samples.

Key idea: To read a parametrization through sample values, substitute each chosen parameter into every component, then treat the resulting coordinates together as one point in space.

Evaluating a Vector-Valued Function

Let \(I\) be an interval of real numbers. A vector-valued function \(\mathbf r:I\to\mathbb R^3\) can be written in component form as

$$ \mathbf r(t)=\langle x(t),y(t),z(t)\rangle. $$

Here \(t\) is the parameter, and \(x(t)\), \(y(t)\), and \(z(t)\) are the three coordinate functions. For any chosen value \(t_0\in I\), its image is the position vector

$$ \mathbf r(t_0)=\langle x(t_0),y(t_0),z(t_0)\rangle. $$

The point has coordinates \((x(t_0),y(t_0),z(t_0))\). The vector and the point have the same three numerical entries, but they play different roles: the vector records position relative to the origin, while the point is the location in space.

Definition: A sample value is a selected parameter value \(t_0\) in the domain of \(\mathbf r\). Its sample point is the point with coordinates \((x(t_0),y(t_0),z(t_0))\).

Evaluation is componentwise: use the same parameter value in all three components. A value that is not in the function's domain cannot be used as a sample. If a component is given by a formula that is defined only for some inputs, the chosen parameter must be allowed by every component.

Why Substitution Works Component by Component

The rule follows directly from what the component form means. The output \(\mathbf r(t)\) is defined to have first coordinate \(x(t)\), second coordinate \(y(t)\), and third coordinate \(z(t)\). Setting \(t=t_0\) therefore sets the input in each coordinate function to \(t_0\), not just in one of them.

1
Start with the component form.
Write \(\mathbf r(t)=\langle x(t),y(t),z(t)\rangle\).
2
Choose an allowed input.
Take \(t_0\in I\), so that the evaluation is defined.
3
Evaluate every coordinate function.
The result is \(\mathbf r(t_0)=\langle x(t_0),y(t_0),z(t_0)\rangle\), which supplies the three coordinates of one point.

This is also a short justification of the evaluation rule: the output is an ordered triple by definition, and substituting \(t_0\) into that triple replaces each component function by its value at \(t_0\). No coordinate can be omitted or evaluated at a different parameter value.

Worked Example: Three Sample Points from a Polynomial Parametrization

Worked Example: Three Sample Points from a Polynomial Parametrization

Let \(\mathbf r(t)=\langle t+1,t^2,3-t\rangle\) for \(-1\le t\le 2\). Find the sample points at \(t=-1\), \(t=0\), and \(t=2\), and describe their order.

At \(t=-1\), substitute \(-1\) into all three components:

$$ \begin{aligned} \mathbf r(-1) &=\langle (-1)+1,(-1)^2,3-(-1)\rangle\\ &=\langle 0,1,4\rangle. \end{aligned} $$

The first sample point is \((0,1,4)\).

At \(t=0\):

$$ \begin{aligned} \mathbf r(0) &=\langle 0+1,0^2,3-0\rangle\\ &=\langle 1,0,3\rangle. \end{aligned} $$

The second sample point is \((1,0,3)\).

At \(t=2\):

$$ \begin{aligned} \mathbf r(2) &=\langle 2+1,2^2,3-2\rangle\\ &=\langle 3,4,1\rangle. \end{aligned} $$

The third sample point is \((3,4,1)\).

Since the selected parameter values increase in the order \(-1,0,2\), the corresponding sampled locations, in that order, are \((0,1,4)\), \((1,0,3)\), and \((3,4,1)\). As a check, substituting each parameter into the original three formulas gives precisely the three displayed coordinate triples.

Keep the Points in Parameter Order

A list of sample points is most useful when it preserves the order of the parameter values used to produce them. For instance, if the chosen values satisfy \(t_1<t_2<t_3\), record the corresponding outputs as \(\mathbf r(t_1)\), \(\mathbf r(t_2)\), and \(\mathbf r(t_3)\), in that order. This tells you which location corresponds to each selected input and gives a sequence of sampled locations.

The parameter values need not be equally spaced. In particular, consecutive entries in a sample list need not represent equal changes in \(t\). Also, equal changes in the parameter do not imply equal changes in the coordinates. The coordinate functions determine the output at each chosen value; the parameter order alone does not provide distances between points or a description of what happens between them.

Important distinction: The parameter \(t\) is the input to the vector-valued function. The coordinates \(x(t)\), \(y(t)\), and \(z(t)\) are the output. In general, \(t\) is not one of the coordinates.

Worked Example: A Coordinate Can Change Direction Across Samples

Worked Example: A Coordinate Can Change Direction Across Samples

Let \(\mathbf r(t)=\langle t^2,t-1,2t+3\rangle\), and use the sample values \(t=-2\), \(t=0\), and \(t=1\).

At \(t=-2\):

$$ \begin{aligned} \mathbf r(-2) &=\langle (-2)^2,(-2)-1,2(-2)+3\rangle\\ &=\langle 4,-3,-1\rangle. \end{aligned} $$

At \(t=0\):

$$ \begin{aligned} \mathbf r(0) &=\langle 0^2,0-1,2(0)+3\rangle\\ &=\langle 0,-1,3\rangle. \end{aligned} $$

At \(t=1\):

$$ \begin{aligned} \mathbf r(1) &=\langle 1^2,1-1,2(1)+3\rangle\\ &=\langle 1,0,5\rangle. \end{aligned} $$

In increasing parameter order, the sampled points are \((4,-3,-1)\), then \((0,-1,3)\), then \((1,0,5)\). The first coordinate changes from \(4\) to \(0\) and then to \(1\); it does not simply increase throughout these samples. The second and third coordinates have their own values and changes. Checking the substitutions in the original component formulas confirms each point. This example shows why a description should be based on the computed coordinates rather than on an assumption that every coordinate changes in the same way.

Sample Points Give Partial Information

Evaluating several parameter values gives reliable information about the locations at those inputs. It does not, on its own, show the complete set of locations produced for all parameter values. Two selected sample points do not specify which points, if any, occur between them, nor do they determine the behavior of the parametrization at unsampled values.

For example, the points at \(t=0\) and \(t=1\) are just the two outputs \(\mathbf r(0)\) and \(\mathbf r(1)\). To say what the function does at \(t=1/2\), evaluate \(\mathbf r(1/2)\); do not infer its coordinates merely by averaging the endpoint coordinates. Such averaging is not a general evaluation rule.

Takeaway: A sample list describes selected outputs, not automatically every point in the curve. Each claim about a location at a particular parameter value should come from evaluating the parametrization at that value.

Worked Example: Samples at Unequally Spaced Values

Worked Example: Samples at Unequally Spaced Values

Let \(\mathbf r(t)=\left\langle \frac{1}{t+1},t^2,t-2\right\rangle\) on \(0\le t\le 2\). Evaluate it at \(t=0\), \(t=1\), and \(t=2\), then arrange the points in parameter order.

The denominator \(t+1\) is nonzero at all three selected values, so these evaluations are defined. At \(t=0\):

$$ \begin{aligned} \mathbf r(0) &=\left\langle \frac{1}{0+1},0^2,0-2\right\rangle\\ &=\langle 1,0,-2\rangle. \end{aligned} $$

At \(t=1\):

$$ \begin{aligned} \mathbf r(1) &=\left\langle \frac{1}{1+1},1^2,1-2\right\rangle\\ &=\left\langle \frac12,1,-1\right\rangle. \end{aligned} $$

At \(t=2\):

$$ \begin{aligned} \mathbf r(2) &=\left\langle \frac{1}{2+1},2^2,2-2\right\rangle\\ &=\left\langle \frac13,4,0\right\rangle. \end{aligned} $$

The values \(0,1,2\) are in increasing order, so the sampled locations are \((1,0,-2)\), \((\frac12,1,-1)\), and \((\frac13,4,0)\). Substitution into the original formula checks all three coordinates in each result. The coordinates are exact fractions; no rounding is needed.

A Common Substitution Error

A common mistake is to substitute the selected value into only one component while leaving the other components unevaluated. For the parametrization \(\mathbf r(t)=\langle t+1,t^2,3-t\rangle\), a mistaken calculation at \(t=2\) might be

$$ \mathbf r(2)=\langle 2+1,t^2,3-t\rangle=\langle 3,t^2,3-t\rangle. $$

This is not a point with numerical coordinates: \(t\) is still present in two entries. The correction is to use \(2\) in every component:

$$ \mathbf r(2)=\langle 2+1,2^2,3-2\rangle=\langle 3,4,1\rangle. $$

Another error is to read the input parameter as though it were automatically a coordinate. In the same example, \(t=2\) produces the point \((3,4,1)\), not the point \((2,2,2)\). Only the component formulas determine the output coordinates.

Check Your Understanding

For each question, keep the parameter value separate from the point it produces.

  1. If \(\mathbf r(t)=\langle t-2,3t,t^2\rangle\), what point is produced at \(t=1\)?
  2. For \(\mathbf r(t)=\langle t^2,1-t,2\rangle\), evaluate the function at \(t=-1\) and \(t=2\), and list the points in increasing parameter order.
  3. Why must the same selected parameter value be substituted into all components?
  4. If you know \(\mathbf r(0)\) and \(\mathbf r(1)\), does that alone determine \(\mathbf r(1/2)\)? Explain.
  5. What information does an ordered list of sample points preserve about the selected parameter values?

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