The Geometer’s Sketchpad was intended primarily as a learning tool, but it also works quite well for drawing illustrations. I have used the graphics extensively in test papers and in my web pages. I have seen textbook images that appear to have been drawn with Sketchpad. The perspective tools certainly can be used for these purposes. First turn the image around until you have a good idea of what view you want. Then change line weights, hide lines, truncate lines, hide points, and construct polygon interiors until you have the look you want.
Below is a globe made with 35 shaded loci. It was not a complicated construction, but it is a rather large file. I like to keep the locus samples setting at the maximum. This results in smoother curves and more gradual shade transition, but it makes the file big and slow. Two views are shown in order to demonstrate one of the possibilities of the shading. The view on the left fades from front to back. For the view on the right, the shade depth was set near zero. Then the shift was adjusted so that the back part of the globe disappears. It is not a perfect world though. The back part is not transparent; it is white. And it is not really in back; it is on the same screen as all of the other points. It overlaps the front part in places, creating a bit of discontinuity in the grid. Click on the image to download the file.
Another interesting effect can be created by making the depth setting negative and very close to zero. Moving the shift in slowly causes layers of the image to be shaved off a little at a time.
The primary purpose of the colored solids is for use in illustrations. They can be used in web pages (as they are in this one), or in printed classroom material, such as tests and other handouts. I have always found Sketchpad very convenient in that regard. Normally it is just a matter of selecting the objects, copying them, and pasting them into a document. That is not possible with traces though. They cannot be selected. Screen capture works well though. A good graphics editing program will facilitate the capture. If that is not available, Windows will copy the screen onto the clipboard onto the clipboard when Alt+[Print Screen] is pushed. If you are working from some other platform, consult a manual.
This is an image of ten spheres stacked in a tetrahedral pattern. The outlines were all hidden. The trace tends to leave rough edges. To soften them, a very large image was captured, then reduced by a factor of 50%. Actually only nine of the spheres were constructed, since the tenth one is completely hidden. The background objects must be painted first, then the foreground. One of the shortcomings can be noticed here. Although the lighting is consistent, the objects do not throw shadows. Otherwise the spheres on the right side would be in the shade of the others. Do not expect that to change in the future. These tools are already more complex than they need to be, and no one is going to use the Geometer's Sketchpad to remake Snow White.
The cone at left is an example of an image that might be used on a test or other printed handout. Few of us use color on material that is to be reproduced in large numbers, so this was done in gray scale. For gray, set all three of the color slides to the same value. The lines were kept for this image. Their colors and weights were changed. The image was reduced by a factor of 30%. If this image were meant for printing, I would create a full-sized image, paste it into a document, and then reduce it there. Notice that the right angle mark is actually acute and the mark on the central angle on the base is elliptical. They were both constructed and plotted in perspective, which enhances the feeling depth.
Changing the colorbar will have no effect on a solid that has already been painted, but if it is painted again, it will have the new color. In this way it is possible to create several solids of different colors using the same colorbar. The color of a solid is determined by the last colorbar that was created before the solid. By creating a new colorbar before each new solid, it is possible to change each of them separately. The three objects below were captured from the same file. Each has its own colorbar. It is also possible to give them each a separate light source by creating several sundials, but that would only serve to make the lighting inconsistent.
The solid constructions are all quite complex. As a consequence, the files will tend to be very large and slow. Bear this fact in mind if size and speed are concerns.
You may have noticed that some of the sample files are very large. Each point created with the perspective tools has twenty or more hidden objects in the background. Depending on your situation, you may have some reason for keeping your files small. This information may help.
The Segment tool creates two endpoints and connects them with a segment. It will create these endpoints (and their ancestors) even if you already have points that are identically defined. Take the example of this icosahedron. Each vertex is an endpoint of five different edges. If all of the edges were created with the Segment tool, then there would be five identical points at each vertex. The most economical way of doing this is to construct only the vertices, one time each. Then draw the segments without using the custom tools. This can be difficult to do if a figure has many vertices. Try drawing it in small sections.
With the shaded segments there is no way around the problem of duplicate points. A new point must be constructed for each end of every shaded line segment. The shaded segments are actually loci.
These controls work for me, but you may have ideas of your own. Most of the input selections are assumed. The Perspective Setup, Shadebar, Sundial, and Colorbar tools create the objects, give them the necessary labels, and hide them. The hidden points can be overridden by constructing new points with the same names. That is how the solids sketch was created. Try it yourself. First construct the tetrahedron. Drag it aside. Create a new point, and label it image center. The label must be shown. Now construct a cube. Its image will be centered on the new point. If the label is hidden, the tools will center the next construction on the earlier point.
Below is an explanation of the function of each object.
|
x-y center |
The x and y coordinates of a pre-image point are measured with respect to this point. |
|
z center |
The z coordinate of a pre-image point is measured with respect to this point. |
|
image center |
This point is the transformation of the origin. The image will rotate about it. |
|
dial |
This is the center of the control dial. |
|
spin |
Rotation about the z-axis |
|
pitch |
Rotation about the x-axis |
|
roll |
Rotation about the y-axis |
|
distance |
This sets the distance from the observer to the origin. When this value is one, the distance is 5 cm. |
|
scale |
When this value is one, all points on the plane of the image center will be full-scale. |
|
perspective toggle |
0 = orthogonal projection 1 = perspective view |
|
shade depth |
This sets the distance between the planes that bookend the shade transition zone. When this value is one, the planes are 5 cm apart. |
|
shade shift |
This moves the shade transition zone in and out. When it has a value of zero, the zone is centered on the image center. The zone will be shifted by 5 cm times this value. |
|
sunx
|
These are the components of a unit vector pointing to the light source, which is a point at infinity. The values are defined by the Sundial tool and used only by the traced solids. |
|
red
|
These are numbers between zero and one. They determine the color value for a point on the surface of a traced solid where the light ray is tangent to the surface. The variable color of the rest of the surface also depends on these values. The values are defined by the Colorbar tool and used only by the traced solids. |
|
sunny
|
These are numbers between zero and one. On the lighted side of a traced solid, the color is gradually moved from the red-green-blue setting (above) toward white; on the shaded side, it moves toward black. These values determine how close the color can get to white or black. For example, if sunny is set to 1, then at a point where the surface is facing the light and normal to the light ray, the color will be completely white. The values are defined by the Colorbar tool and used only by the traced solids. |
When creating these tools, I was careful to avoid the commands that are not yet supported by Java Sketchpad. All of the simple constructions can be translated to Java, however the view buttons are presentation buttons and are not yet supported. If you wish to offer different types of views in the same sketch, custom controls will have to be constructed. Java Sketchpad also does not support shading. Shaded objects and parametrically colored objects will not appear in the Java version. Refer to the tips on conserving memory. Some of these larger files would be completely out of hand if they were converted to Java.
Back to Perspective Tools Manual
Back to Whistler Alley Mathematics