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Section 2.2 Space Curves

Examples:

See the list of example Space Curves on the Examples submenu of the CalcPlot3D main menu. See more about the Examples menu in Section 4.6.

To add a space curve to the plot, select the option Space Curve: r(t) on the Add to graph drop-down menu, located just below the control panel toolbar buttons. You'll see an object dialog appear like the following:

Space curve object
Helix space curve
Figure 2.2.1 Space Curve dialog
Figure 2.2.2 Helix space curve

Now let's plot the default space curve.

Clear Plot button

Clear the plot by clicking the clear plot button.

Then check the checkbox in the upper-left corner of this object to plot the default curve.

A multicolored helix should appear on the 3D plot as shown in Figure 2.2.2. The color variation along the curve is meant to help viewers see the curve’s 3D features, clearly showing which parts of the curve are in front and which are in back.

Space curves can also benefit from the other 3D modes, but when using 3D glasses modes, it is helpful to choose the option Use Constant Primary Color. The constant color is automatically selected when in 2D mode.

If you wish to view the a 2D plane curve on the \(xy\)-plane, select the Restrict view to 2D checkbox.

The range for \(t\) and the number of steps can each be adjusted. More steps may be needed to make the curve smoother, but the number of steps will also affect the speed of the animation of motion along the curve.

To animate the motion along the curve, click the Animate button. This will cause a blue trace vector to move along the curve from minimum \(t\)-value to maximum \(t\)-value. By default the velocity vector (black) and acceleration vector (green) are also displayed as the point moves along the space curve.

The slider or textbox can also be used to adjust the value of \(t\) and the position of the object on the curve.

Subsection 2.2.1 Space Curve Settings, ⚙

Clicking on the settings button ⚙ will open up the following menu.

Space Curve settings
Show Trace Point:

Indicates whether to display a point at the current position on the curve. Selected by default.

Show Trace Vector:

Indicates whether to display the blue trace vector at the current position on the curve. Selected by default.

Show Velocity Vector:

Indicates whether to display black velocity vector at the current position on the curve. Selected by default.

Show Acceleration Vector:

Indicates whether to display the green acceleration vector at the current position on the curve. Selected by default.

Show Osculating Circle:

Indicates whether to display the osculating circle at the current position on the curve.

Show Curvature Value:

Indicates whether to display the curvature value at the current position on the curve.

Show whole TNB-Frame:

Indicates whether to display the entire TNB-Frame at the current position on the curve. Selecting or deselecting this option will automatically select or deselect all the TNB-Frame options shown below.

TNB-Frame Options:

Indicates whether to display individual parts of the TNB-Frame at the current position on the curve. You can either choose to select just a few items to begin or select the whole TNB-Frame above and then deselect a few items.

Animation Settings:
  • Continuous Animation - If selected, the animation will continue until stopped by the user.
  • Bounce over range - If selected along with continuous animation, the animation will go through the range of \(t\) values in reverse after reaching the maximum \(t\) value, and then reverse again once it reaches the minimum \(t\) value.