Lab 3: Ellipses and Axonometry

Activity 1| Activity 2 | Activity 3 | Activity 4

Activity 1: Axonometry: intersection of a cylinder and a plane

We will draw a figure showing the intersection of a right circular cylinder and a plane, in a view using the aerial projection.

Step 1.  Set up the 3 coordinate axes.

• In a new sketch, choose Define Coordinate System and Hide Grid.  Rotate the unit point on the x-axis by +90 degrees, also by –30 degrees, and -120 degrees to create the points E3, E1, E2. Let the origin be labeled O.
• Hide the x-axis and its unit point.
• Draw the lines OE1 and OE2. You can either use the y-axis or draw a line OE3.

Step 2.  Set up a plane

• Construct points A1, A2, A3, on the respective axes. 
• Then draw 3 lines through the pairs of points to give an extended triangle.  The triangle lies in the plane e of the 3 points. Use color as needed.
• You may also draw the 3 segments and make them thick for visual clarity. 
• Then draw a point P, which we will think of as a point in the OE1E2 plane.

Step 3: Construct the projection of P on the plane e

• Construct lines h (horizontal) and v (vertical) through P parallel to OE1 and OE3. 
• Intersect the line h with line A1A2 to get Q. 
• Construct the line s through Q parallel to A1A3.  Intersect s with line v to get P'.
• Construct the segment PP' and make it thick.
• Now P' is the accurate picture of the projection of P on plane e.  The vertical distance PP" is the actual distance from the (x,y) plane of point P'.
• Drag P around and see that when P is near one of the points O, A1 or A2, that P' is near A3, A1 or A2.

Step 4. Projecting a circle and other shapes

• Hide the 3 lines through PQP'. Draw a circle. Select point P and the circle and Choose Merge Point to Circle from the Edit Menu.
• Select P and P' and choose Locus from the Construct Menu.
• Now Select P and Choose Animate Point to see how the segments move up and down as P moves around the circle.  Move the circle around to see different ellipses.
• Also, you can select the segment PP' and choose Locus.  But you should use Properties to reduce the number of segments being plotted to about 12 in order to see the segments clearly.

Step 5. Project Other Shapes

• You can project other shapes by drawing a shape such as a polygon interior.
• Split P from the circle and merge with the new shape.

Activity 2: Axonomic Picture of a Line PQ in space

Part 1: Draw axes as before.
Part 2. Draw points and line P.

• Draw points P and Q.  These are views of lines in space.  To show their z-coordinates, construct lines p and q through P and Q parallel to OE3.
• Construct point P1on p and Q1 on q.  P1 and Q1 are the projections of P and Q to the OE1E2 plane.
• Construct segments PP1 and QQ1.
• Construct line PQ.  This is a view of a line in space.
• Construct line P1Q1.  This is a line in the OE1E2 horizontal plane.
• Let the intersection of the two lines be R.  This shows the intersection of line PQ with the plane OE1E2.

Part 3. Traces of line PQ on the vertical coordinate planes.

It gives a clearer view of the line PQ to construct its intersections with the vertical coordinate planes.

• Intersect line P1Q1 with the lines OE1 and OE2.  Then construct the vertical lines through these intersection points parallel to OE3. 
• Intersect the two vertical lines with line PQ to get the intersection of line PQ with the planes OE1E3 and OE2E3.

Part 4. Coordinates of P and Q

• Figure out how to measure the x, y, z coordinates of P and Q in the figure above and carry out the calculations.
• Also, measure the coordinates of the intersection points of P and Q with the vertical coordinate planes.
• Check your work by taking P = (1, 1.2, 1) and Q = (.8, 2, 2.2) and calculate exactly by hand the coordinates of the intersections of line PQ with the vertical coordinate planes.  Compare with your Sketchpad measurements.

Activity 3: Ellipses as images of circles

• Start with a circle and perpendicular lines as in the figure.
• Construct a point P on the circle with center O through X, and from P construct the points P1 and P2 as shown.  You should be able to drag P around the circle without changing the circle.
• The measured ratios are coordinates of P. 
• Now draw a triangle A0A1A2 and construct P' with the same affine coordinates relative to this triangle:  With center A0, dilate A1 by ratio OP1/OX to get P1' and dilate A2 by the other ratio to get P2'.  Than translate P2' by A0P1' to get P'.

• Select P and P' and choose locus to see the ellipse that is the image of the circle.

Activity 4: Dandelin Spheres in a Cylinder

Reproduce with Sketchpad the figure with two parallel lines and two circles tangent to the lines from the handout in class.

• Specifically, start with two parallel lines and points E1 and E2 and construct the rest.
• Then elsewhere in the sketch, construct a rectangle whose width is the diameter of the circles and whose length equals E1E2.
• Then construct an ellipse inside the rectangle with the major and minor axis equal to the length and width of the rectangle.  Do this by constructing first a circle centered at the center of the rectangle with diameter equal the width.  Then "stretch" the circle to a locus using the ideas above (this will be simpler).