## Parametrised curves in space

Content: This page contains an applet for visualising spatial curves and instructions for its use.

Applet

Theory

Analysis for Computer Scientists, Chapter 14

Help

### Entering a parametrised curve

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

common factor (x-component, y-component, z-component)

or

(x-component, y-component, z-component),

where the factor and the components are expressions in the variable t. For example, the screenshot shows the helix given by (cos(t),sin(t),t). 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, where the x-axis is in horizontal direction, the z-axis in vertical direction, and the (positive) y-axis points towards the eye. ### 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 binormal 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.

### Projection

The tab Projection gives the option to control the mapping of the surface onto the plane of projection. With the options Perspective and Parallel projection you can choose between these two types of projection. If perspective is selected the distance of the eye from the surface can be adjusted by the minus and plus buttons. The combo box Projection plane allows one to specify a parallel projection onto one of the coordinate planes. The button Reset restores the default values.

### Rotation of the curve

By moving the mouse while pressing the left mouse button the curve can be rotated.

### Scaling of the axes, boxes and labels

The tab Axes allows one to control the scaling of the axes uniformly or for every axis independently. Note that the independent scaling of the axes is deactivated if the moving frame is drawn. In the field Axes model it is specified if a box is drawn and the axes are labelled.

### 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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