Parametric curves
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01 Theory
Theory 1
Parametric curves are curves traced by the path of a ‘moving’ point. An independent parameter, such as
for ‘time’, controls both and values through Cartesian coordinate functions and . The coordinates of the moving point are . Parametric curve
A parametric curve is a function from parameter space
to the plane given in terms of coordinate functions: Other notations
Be aware that sometimes the coordinate functions are written with
and (or yet other letters) like this: Or simply equating coordinate letters with functions:
Sometimes a different parameter is used, like
or . For example, suppose:
The curve traced out is a parabola that opens horizontally:
Given a parametric curve, we can create an equation satisfied by
and variables by solving for in either coordinate function (inverting either or ) and plugging the result into the other function. In the example:
This is the equation of a parabola centered at
that opens to the right. Image of a parametric curve
The image of a parametric curve is the set of output points
that are traversed by the moving point. A parametric curve has hidden information that isn’t contained in the image:
- The time values
when the moving point is found in various locations. - The speed at which the curve is traversed.
- The direction in which the curve is traversed.
We can reparametrize a parametric curve to use a different parameter or different coordinate functions while leaving the image unchanged.
In the previous example, shift
by : Since the parameter
Link to originaland the parameter both cover the same values for , the same curve is traversed. But the moving point in the second, shifted version reaches any given location one unit earlier in time. (When in the second version, the input to and is the same as when in the first one.)
02 Illustration
Example - Parametric circles
Parametric circles
The standard equation of a circle of radius
centered at the point : This equation says that the distance from a point
on the circle to the center point equals . This fact defines the circle. Parametric coordinates for the circle:
For example, the unit circle
Link to originalis parametrized by and .
Example - Parametric lines
Parametric lines
(1) Parametric coordinate functions for a line:
Compare this to the graph of linear function:
Vertical lines cannot be described as the graph of a function. We must use
.
(2) Parametric lines can describe all lines equally well, including horizontal and vertical lines.
A vertical line
is achieved by setting and . A horizontal line
is achieved by setting and . A non-vertical line
may be achieved by setting and , and .
(3) Assuming that
, the parametric coordinate functions describe a line satisfying: and therefore the slope is
and the -intercept is .
(4) The point-slope construction of a line has a parametric analogue:
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Example - Parametric ellipses
Parametric ellipses
The general equation of an ellipse centered at
with half-axes and is: This equation represents a stretched unit circle:
- by
in the -axis - by
in the -axis Parametric coordinate functions for the general ellipse:
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Example - Parametric cycloids
Parametric cycloids
The cycloid is the curve traced by a pen attached to the rim of a wheel as it rolls.
It is easy to describe the cycloid parametrically. Consider the geometry of the situation:
The center
of the wheel is moving rightwards at a constant speed of , so its position is . The angle is revolving at the same constant rate of (in radians) because the radius is . The triangle shown has base
, so the coordinate is . The coordinate is . So the coordinates of the point
are given parametrically by: If the circle has another radius, say
, then the parametric formulas change to: Link to original
Calculus with parametric curves
03 Theory - Slope, concavity
Theory 1
We can use
and data to compute the slope of a parametric curve in terms of . Slope formula
Given a parametric curve
, its slope satisfies: Concavity formula
Given a parametric curve
, its concavity satisfies the formula:
Extra - Derivation of slope and concavity formulas
For both derivations, it is necessary to view
as a function of through the inverse parameter function. For example if is the parametrization, then is the inverse parameter function. We will need the derivative
in terms of . For this we use the formula for derivative of inverse functions: Given all this, both formulas are simple applications of the chain rule.
For the slope:
For the concavity:
(In the second step we inserted the formula for
from the slope.)
Pure vertical, Pure horizontal movement
In view of the formula
, we see:
- Pure vertical: when
and yet - Pure horizontal: when
and yet When
Link to originalfor the same , we have a stationary point, which might subsequently progress into pure vertical, pure horizontal, or neither.
04 Illustration
Example - Tangent to a cycloid
Tangent to a cycloid
Find the tangent line (described parametrically) to the cycloid
when . Solution
(1) Compute
and . Find
and :
(2) Plug in
:
(3) Apply formula:
: Calculate
at : So:
This is the slope
for our line.
(4) Need the point
for our line. Find at . Plug
into parametric formulas:
(5) Point-slope formulation of tangent line:
Inserting data:
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Example - Vertical and horizontal tangents of the circle
Vertical and horizontal tangents of the circle
Consider the circle parametrized by
and . Find the points where the tangent lines are vertical or horizontal. Solution
(1) For the points with vertical tangent line, we find where the moving point has
(purely vertical motion): The moving point is at
when , and at when .
(2) For the points with horizontal tangent line, we find where the moving point has
(purely horizontal motion): The moving point is at
Link to originalwhen , and at when .
Example - Finding the point with specified slope
Finding the point with specified slope
Consider the parametric curve given by
. Find the point where the slope of the tangent line to this curve equals 5. Solution
(1) Compute the derivatives:
Therefore the slope of the tangent line, in terms of
:
(2) Set up equation:
(3) Find the point:
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05 Theory - Arclength
Theory 2
Arclength formula
The arclength of a parametric curve with coordinate functions
and is: This formula assumes the curve is traversed one time as
increases from to . Counts total traversal
This formula applies when the curve image is traversed one time by the moving point.
Sometimes a parametric curve traverses its image with repetitions. The arclength formula would add length from each repetition!
Link to originalExtra - Derivation of arclength formula
The arclength of a parametric curve is calculated by integrating the infinitesimal arc element:
In order to integrate
in the variable, as we must for parametric curves, we convert to a function of : So we obtain
and the arclength formula follows from this:
06 Illustration
Example - Perimeter of a circles
Perimeter of a circle
(1) The perimeter of the circle
is easily found. We have , and therefore:
(2) Integrate around the circle:
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Example - Perimeter of an asteroid
Perimeter of an asteroid
Find the perimeter length of the ‘asteroid’ given parametrically by
for .
Solution
(1) Notice: Throughout this problem we use the parameter
instead of . This does not mean we are using polar coordinates! Compute the derivatives in
:
(2) Compute the infinitesimal arc element.
Plug into the arc element, simplify:
(3) Bounds of integration?
Easiest to use
. This covers one edge of the asteroid. Then multiply by 4 for the final answer. On the interval
, the factor is positive. So we can drop the absolute value and integrate directly. Absolute values matter!
If we tried to integrate on the whole range
, then really does change sign. To perform integration properly with these absolute values, we’d need to convert to a piecewise function by adding appropriate minus signs.
(4) Integrate the arc element:
Finally, multiply by 4 to get the total perimeter:
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