Intersection of 3D surfaces with cutting planes

While a VRML browser gives the user the flexibility to examine a 3D display from an arbitrary direction, we also wanted to give the user a greater ability to interact with the visualization of the functions. DLMF project members noted that one very useful feature would be the ability to study the changes that occur in the intersection of a plane with a surface as the plane is moved through the surface. Using VRML we were able to create files that would generate cutting planes through a 3D surface. However, currently, the cutting planes are limited to planes perpendicular to the X and Y coordinate axes.

We defined a VRML PROTO called Cutplane to display the plane and compute the intersection of the plane with the surface data. In VRML a PROTO is a user defined reusable component. Cutplane searches the data to determine which surface points are closest to the specified plane and then performs a linear interpolation to obtain the coordinates for the intersection.

When the user opens the VRML display, the 3D-surface and a control panel appear. Figure 4 shows the look of the VRML display when the VRML browser CosmoPlayer is used. CosmoPlayer can be downloaded from http://cosmosoftware.com. We defined another VRML prototype, PROTO Dash to deal with the operation of the control panel. On the control panel, there are two lines of buttons. When the user clicks the X or Y button on the top line, the specified cutting plane will be shown. For example, clicking the X button causes a bounding box to appear around the figure and produces a cutting plane that will move perpendicular to the X axis. The buttons on the second row control the animation similar to those on a VCR: rewind, reverse, stop, play, and fast forward. When the user clicks the VCR buttons, the cutting plane moves in sync with the projected intersection curve, which is displayed on opposite faces of the bounding box. The text on the bottom of the browser shows the value of the cutting plane location.

Figure 4: VRML display on CosmoPlayer.

Figure 5 shows the imaginary part of Airy function $\vert\mbox{Ai}'(z)\vert$ with a cutting plane in the Y direction. Figure 6 presents the real part of Airy function Bi($z$) with a cutting plane in the X direction. These and other figures may be found at http://zing.ncsl.nist.gov/~wang/mathlib/new_ui/.

Figure 5: VRML display with Y direction cutting plane.
Figure 6: VRML display with X direction cutting plane.