Transformation Geometry -- Math 331

February 9, 2004

Discussion

Exercises due Wednesday, February 11

  1. Let A, B, and C be the points

      A  =  (0, -1)  ,   B  =  (3, 4)  ,   C  =  (-1, 1)    . 

    1. Find the point P where the three altitudes of DeltaABC meet.

    2. Find the areas of the three triangles: DeltaBCP, DeltaCAP, and DeltaABP.

    3. Find a triple of homogeneous coordinates for P relative to A, B, and C.

  2. Show that three distinct points A, B, and C are collinear if there is a triple of numbers (u, v, w), not all zero, of weight 0, i.e., u + v + w = 0, such that uA + vB + wC = 0.

  3. Let f(x) = Ax be the linear transformation of the plane where A is the matrix

     A   =   {1}/{5}
    (
    4
    -4
    3
    )
        . 

    1. What points x of the plane are “fixed” by f, i.e., satisfy f(x) = x ?

    2. What lines in the plane are carried by f to other lines?

    3. What lines L in the plane are “stabilized” by f, i.e., satisfy the condition that f(x) is on L if x is on L ?

  4. Find homogeneous coordinates relative to the vertices of a given triangle for the point where the three perpendicular bisectors of the sides of the triangle meet.

    Hint: Use the fact that the perpendicular bisectors are the altitudes of the triangle whose vertices are their feet (i.e., the midpoints of the three sides).


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