## Parametrised curves in plane

Content: This page contains an applet for the visualisation of planar curves and instructions for its use.

Applet

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Theory

Analysis for Computer Scientists, Chapter 14

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### Entering a parametrised curve

As can be seen from the screenshot, the curve must be defined in the field (x(t),y(t)). It is entered in the form

common factor (x-component, y-component)

or

(x-component, y-component),

where the factor and the components are expressions in the variable t. For example, the screenshot shows the spirale given by t(cos(t),sin(t)). For the input the syntax t*(cos(t),sin(t)) or (t*cos(t),t*sin(t)) is also possible. Detailed information on the syntax can be found here.

In the field Interval the parameter interval is entered in the format [lower boundary, upper boundary]. Pressing the button Compute displays the curve on the screen. ### Determining the resolution

In our visualisation the curve is approximated by a polygon. Its number of vertices can be adjusted manually by using the slider if the checkbox Adjust number of points manually is active.

### Velocity vector, acceleration vector, and moving frame

After the curve has been drawn the following objects can be computed (provided the curve is differentiable, see below):

• the velocity vector;
• the velocity vector (blue) and the acceleration vector (green);
• the moving frame.

If the desired option is selected, the vector(s) can either be moved along the curve with the slider or by entering the desired value of the parameter into the field Draw at t= . In the status bar, depending on the selected option, the velocity vector (denoted by v), the acceleration vector (denoted by a), the normal vector (denoted by n) and the current value of the parameter are given. Note that the moving frame is scaled for better visualisation. If the number of points is manually adjusted, these vectors are not drawn.

### Numerical differentiation

To determine the velocity vector, the acceleration vector and the moving frame numerical differentiation is employed (i.e., the derivatives are approximated by difference quotients). By comparing the forward and backward differences with the central differences the applet tries to determine if the curve is differentiable at a given point. A message is generated if this test fails. Numerical differentiation is limited by machine precision. Therefore, differentiability might not be correctly determined for "badly scaled" curves.

### Scaling of the axes

After the curve has been drawn, a number of options are available to change the scaling of the axes. The option Scale uniformly uses the same scale on the x- and y-axis. This ensures that a circle appears as a circle and that a right angle (of the moving frame) is drawn as a right angle. The option Scale independently ensures that both axes are adjusted so that the curve fits into the window. Both options also allow for manual adjustment of the axes by using the "+" and "-" buttons.

### Scale coordinate system independently

The section Enlarge independently allows one to scale the four different axes independently. This option is primarily indented to adjust the coordinate system in such a way as to ensure that the velocity and the acceleration vector are completely visible on the screen.

### Questions

If you have further questions or comments, or if you found a bug, please send us an e-mail.

Financially supported by

University of Innsbruck: New Media and Learning Technologies
Austrian Federal Ministry of Education, Science and Research

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