Assignment 5

by:

Amanda Avery

Whitney Burton

Samuel Obara

Cartesian and Minkowski Geometry

(b) What are circles in Minkowski geometry? Explain how to construct the Minkowski circle with center A and radius r using the GSP Construct menu.

o The circles in Minkowski geometry are actually squares in Cartesian geometry with the diagonals parallel to the two coordinate axes.

o Construction: Let A be a point (x, y) in the x-y Cartesian plane (from the graph menu, show axes). Now, the real number r, can be constructed by creating a ray starting at the origin and extending along a positive axis. Then, create a point, r, on the x-axis. The segment from the origin to the point r will have the length of r, our radius. Now, for the Minkowski circle: create the circle with center A and radius length r. Create the lines through A and parallel to both coordinate axes. The segments created from the intersection points of these lines with the circle will create the Minkowski circle. See diagram, on gsp:

 

 

(c) We found three different cases where the equidistant set C in Minkowski geometry have different constructions.

1. If the segments of the square created from the diagonal of AB are parallel to the coordinate axes, then we have the following set C.

Construction:

Given points A and B such that the segments of the square created from the diagonal AB are parallel to the coordinate axes, we can find a set C, such that d(A,C)=d(B,C) in the following sense:

To create a square from the diagonal AB:
Find the midpoint E of segment AB. Create k, the line perpendicular to AB passing through the point E. Create c1, the circle with center E, passing through the point B. Create F and G, the intersections of line k and circle c1. The figure AFBG is the square.

To consruct the equidistant set C:
Create the segment GF. Create s and q, lines parallel to the y-axis, passing through the points G and F, respectively. Create u and w, the lines parallel to the x-axis through the points G and F, respectively. Create c2, the circle with center F and passing through the point E. Construct H and J, the intersection of circle c2 and lines q and u, respectively. Also, create c3, the circle with center G and passing through the point E. Construct K and I, the intersection of circle c3 and lines w and s, respectively. Now, create the following rays: FH, FJ, GK, GI. Now, the set C is made up of the points on the segment FG, the points on the rays FH, FJ, GK, and HI, the set of points contained in the area bounded by the rays FH and FJ, and the set of points contained in the area bounded by the rays GK and GI. ... You can test this by moving the point C1, C2, C3 and C4. See diagram, on gsp:

 

2. If A[Y] >B[Y], and the segments of the square created from the diagonal AB are not parallel to the coordinate axes, then we have the following equidistant set C.

Construction:

With the given points A and B such that A[Y] >B[Y], and the segments of the square created from the diagonal AB are not parallel to the coordinate axes, we can construct the equidistant set C, such that d(A,C) = d(B,C).

To create a square from the diagonal AB:
Find the midpoint E of segment AB. Create k, the line perpendicular to AB passing through the point E. Create c1, the circle with center E, passing through the point B. Create F and G, the intersections of line k and circle c1. The figure AFBG is the square.

To construct the equidistant set C:
Create k and l, the lines parallel to the x-axis passing through the points A and B, respectively. Create m and n, the lines parallel to the y-axis passing through the points G and F, respectively. Create P, the intersection of line k and line n. Also, create Q, the intersection of line l and line m. Create the segment PQ; create the rays PF and QG. The equidistant set C is the points contained on the segment PQ and on the rays PF and QG. ... You can test R and S, to see this. :-) See diagram, on gsp:

3. If A[y]=B1[y] or if A[x]=B2[x], then we have the following equidistant sets.

Construction:

Given two points A and B such that A[y] = B1[y], the equidistant set C is constructed by creating the perpendicular bisector of segment A-B1. The same is also true for A[x] = B2[x]. You can test this by moving the points C1 and C2. See diagram, on gsp:

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