Math 487 Winter 2003: Lab 1

Lab Outline

The lab has 3 parts.  In response to requests, the tasks will be less step-by-step.  We start with an outline.  This is followed by a bit more details, links to techniques, hints, etc.

Part 1. How the signed ratio is basic in Sketchpad's dynamic geometry

This part consists of a couple of short experiments that show how the signed ratio (the affine ratio) is basic in Sketchpad.

Part 2.  Images of Affine Transformations.

Construction Problem

Given a quadrilateral ABCD and a triangle A'B'C', there is an affine transformation T so that the images of A, B, C are A', B', C'.  Construct point D' so that the image of D is D'

Conclusion:  Given two triangles ABC and A'B'C', there is exactly one affine transformation T so that the images of A, B, C are A', B', C'.

Proof: By the construction, for any point D, the image D' is determined by ABC and A'B'C'.

Part 3.  Ratios, lines, and Bezier Curves

Construction Exercise

Construct a curve by "curve stitching" from a "control polygon".  See the directions for details.

Conclusion:  If the control polygon is mapped by an affine transformation, the curve will follow along, i.e., the curve from the new control polygon will be the image of the original curve.

Part 4.  Introduction to Affine Coordinates

This part will be an introduction, as time permits.

Experiments

Given a triangle ABC and a point D, there are a number of ratios that can be associated with D.  Explore these ratios and their relationships.

Conclusion: The point D can be located in the plane relative to ABC in several ways, each of which is important in certain contexts.

Background: Affine Transformations of a Plane

We have been studying parallel projections.  We can use parallel projections to create transformations of a particular given plane p by projecting p to another plane, and then composing with one or more other parallel projections that end by projecting back into p.

The name of such a transformation is an affine transformation of the plane p.

We have seen that a parallel projection of one plane onto another has 3 special properties:

• The image of a line is a line.
• The image of a pair of parallel lines is a pair of parallel lines.
• Given three points A, B, C on a line, if the images are A', B', C, then the signed ratio AC/AB = A'C'/A'B'.

Since an affine transformation is a composition of such parallel projections, an affine transformation has the same 3 properties (check this).

IMPORTANT FACT:  We will show (but have not yet shown) that for any two triangles ABC and DEF, there is an affine transformation T so that the image T(ABC) is DEF. This means that if we draw any two triangles with Sketchpad, there is an affine transformation that takes ABC to DEF.  In the course of this lab, we will see why there is only one such affine transformation.

Part 1. How the signed ratio is basic in Sketchpad's dynamic geometry

• Draw a line AB and construct a point C on the line. Now drag point A or B. Then C must move also.  Notice how C moves as A or B moves. Where does Sketchpad place C?  How would you describe it? 
• If the figures below show first the figure with A, B, C and then with A, B but C missing, where will C be (sketch approximately).  Where is C exactly (tell)?

• Measure a ratio. Select (in order) A, B, and C.  Choose Ratio from the Measure menu.  Check that the ratio on the screen is AC/AB.  (If not, you chose the points in a different order.) Now drag A and B again and see what happens to the ratio.  Then drag C and see where C must be for the ratio to be +2 and also –3, etc.

Sketchpad Method: How to construct a point C with a given ratio AC/AB

In the same sketch, draw a line DE.  There are a couple of ways to construct a point F on DE so that DF/DE = AC/AB.  The simplest way uses the built-in dilation transformation.

• Select the ratio (i.e., the text on the screen) and choose Transform->Mark Scale Factor.
• Select the point D and choose Transform->Mark Center.
• Select point E and choose Transform->Dilate (you will be asked whether you want to use the marked scale factor; you do).
• The new point E' is the desired point F.  Measure DF/DE and check that it is always the same as AC/AB no matter where you drag various points.  Include cases where C is dragged so that the ratio is negative.

Shortcuts:

• You can mark the ratio AC/AB directly without measuring it first.  Just select A, B, C and choose Transform-> Mark Ratio.
• You can mark a center point by double clicking on the point.

Practice Exercise

• Draw a triangle ABC and a second triangle DEF.  By the remark above, we know there is an affine transformation T that maps ABC to DEF.
• Construct a point P on side BC.  Then use the Sketchpad construction above to construct the image Q = T(P) on side EF.  The ratios PB/PC should be the same as QE/QF wherever any of the points are dragged.

Part 2: Image of a Quadrilateral

Construction Problem

Given a quadrilateral ABCD and a triangle A'B'C', there is an affine transformation T so that the images of A, B, C are A', B', C'.  Construct point D' so that the image of D is D'

Hints: You can put more lines into your ABCD figure to get a better handle on it. Construct the diagonals AC anc BD, intersecting at E. Can you construct E', the image of E?

Question: Does your construction work for any D, or are there special cases?

Check the Answer: Compare your results with this web page with the same figure.