Flat view (no strings):Frontal view (strings are dashed lines extending to point P):Java-file at www.mste.uiuc.edu/dildine/classes/ci499fall04/onepoint.htmlJava-file at www.mste.uiuc.edu/dildine/classes/ci499fall04/twopoint.htmlWRAP – UP OF PERSPECTIVE & PROJECTIVE GEOMETRYLinks:A complete GSP file for download to view my sketches and activities.References:James P. Dildine - 1 -CI 499 Fall 2004 - ProjectCOSTRUCTING A PHYSICAL MODEL OF DESARGUES’ THEOREM DAY1Desargues Theorem states the following: Two (non-coplanar) projective triangles are perspective with respect to a point if and only if they are perspective with respect to a line. (Wallace & West p. 360)Here are two possible ways to introduce Desargues’ Theorem:1. Cut a triangle out of an index card. Hold it about arm’s length. Shine a flashlight onto the triangle and notice the projection on a nearby wall. The projection of the triangle is similar to the card-cut-out. Notice how the projection changes based on the relationship of the triangle’s distance from the light, from the wall, and angled from the wall. Each triangle you see is a projection of the one you are holding. 2. Next, Take a piece of twine and cut three equal lengths (about a foot each). Take a file folder and draw a triangle near the center of the inside of the file folder. Now this is tricky because we want to rig our file folder so that the sides will all extend to the folder’s fold line. Label these vertices (A,B,C) Fold the file folder and draw another triangle on the opposite folder flap (on the outside) that appears inside the larger, triangle. Again try and rig your triangle o the sides extend to thefold. Label these vertices (A’, B’, C’) Poke holes into the outside triangle and thread your twine through. Tape the twine pieces to the vertices of the larger triangle. 3. Now, with a partner, one person should hold the pieces of twine together outside of the file folder while the other person opens the file folder and looks into it to see the following. The twine should be taut without too much bending.Two views of how it should look are shown below: P is the point where the person outside the folder should be holding the twine. Note the bottom panel of the file folder should remain flat on your table surface. It might help to tape it down.Two side views:kA'B'C'CABPPBACC'A'B'4. Triangle ABC is considered the projection of triangle A’B’C’. If you are lucky you will also notice the following relationship. Draw a connecting line outwardJames P. Dildine - 2 -CI 499 Fall 2004 - Projectfrom CA to the fold line of the folder, do the same with BA, and BC. Repeat the procedure with A’C’, A’B’, & B’C’. If all goes well you should see something representing the following figures or close to it. It’s hard to keep everything on the file folder but the next activity might help you see the relationship. Flat view (no strings):A'B'C'ABCI NSI DEOUTSI DEYXZB'A'C'ABCFrontal view (strings are dashed lines extending to point P):ZYXPB'A'C'ACB5. Two activities to try: First, take a sheet of paper and attempt to draw in 2-D what we have just represented in 3-D. Second, attempt a GSP construction of the same figure.James P. Dildine - 3 -CI 499 Fall 2004 - ProjectA MODEL OF DESARGUES’ THEOREM AND PERSPECTIVE DRAWING DAY 2Follow these directions to create a Desargues’ Theorem Model. Can be done on paper or GSP. 1. Draw three intersecting lines and label the intersection point P. 2. Inscribe two triangles on these lines and label them ABC, and A’B’C’.As Shown:DCC'A'B'BA3. Extend Corresponding sides (AB, A’B’) Construct their intersection point (label it X). Repeat for Sides BC & B’C’ labeling the intersection Y, and finally AC & A’C’, labeling this last intersection Z. Now, draw a line through the three points. It should look similar to the following picture.AB a nd A'B' - XA C a nd A'C' - YBC a nd B'C' - ZPCC'A 'B 'BAThis is an illustration of Desargues’ Theorem. Restated, we can see the following relationships. Two (non-coplanar) projective triangles are perspective with respect to a point if and only if they are perspective with respect to a line. (Wallace & West p. 360) The two triangles are in perspective through point P. As the smaller triangle isJames P. Dildine - 4 -CI 499 Fall 2004 - Projectprojected onto ABC, looking through point P. In addition, the triangles are in perspective on their planes according to line the XYZ. This figure a nice example to view in Geometer’s Sketchpad as it offers the student the opportunity to examine how the relationships are retained as the figure is moved about the sketch. Try out a decent Java-sketch located online at www.mste.uiuc.edu/dildine/classes/ci499fall04/desargues.htmlJames P. Dildine - 5 -CI 499 Fall 2004 - ProjectONE AND TWO POINT PERSPECTIVE DRAWINGDAY 2Directions for One-point are given here (we’ll create a prism).The easiest way to start is to define your vanishing point. Create a point anywhere onthe sketch. Then create a triangle below this point. Connect each of the vertices to the vanishing point. Create a point somewhere on any segmentThrough that point and using an adjacent edge create a parallel line and the intersection point. Repeat for the next adjacent side. Giving you a figure like this:Now clean it up and connect vertices of the triangles that will appear as the following:Clean up your sketch and manipulate your figure to view your prism in perspective. Java-file at www.mste.uiuc.edu/dildine/classes/ci499fall04/onepoint.htmlDirections for Two-point perspective drawing (this time a box).James P. Dildine - 6 -CI 499 Fall 2004 - ProjectCreate a horizontal segment that will serve as our horizon. And create the front edge of our box. Horizon Lin eConnect the points on the front of our box as shown:Horizon LineCreate points on the top segments of our box and while they are selected create parallel lines to the front of our box.James P. Dildine - 7 -CI 499 Fall 2004 - ProjectHorizon LineFinally connect the intersections and the horizon points as shown:Horizon LineClean up your sketch and manipulate your figure to view your box in perspective. Java-file at www.mste.uiuc.edu/dildine/classes/ci499fall04/twopoint.htmlJames P. Dildine - 8 -CI 499 Fall 2004 - ProjectWRAP – UP OF PERSPECTIVE & PROJECTIVE GEOMETRY- A Flashlight is not the best way to do this! Why? Neither is an overhead projector…Why? -