MATD02 -Fall 2022 Assignment #2

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MATD02 -Fall 2022

Assignment #2

Due to Monday November 14

1.   Show that on a sphere of radius R, the area of a spherical circle of radius r is A = 4几R2 sin2 ( ).

2.   On the unit sphere, f(x) = ( x, y, z) defines an isometry. Why? What are the fixed points on this map?

In hyperbolic plane, a Saccheri quadrilateral is a quadrilateral with a pair of congruent    opposite sides that are both perpendicular to a third side, and a Lambert quadrilateral is a quadrilateral with three right angles. For a Saccheri quadrilateral, we make the following definitions:

• The two congruent opposite sides are called the legs.

• The side perpendicular to the legs is called the base.

• The side opposite the base is called the summit.

• The two angles adjacent to the summit are called the summit angles. 

• The segment joining the midpoints of the summit and the base is called the midsegment.

In a Lambert quadrilateral:

• The angle that is not a right angle is called the fourth angle.

• The vertex of the fourth angle is called the fourth vertex.

3.         Prove that every Saccheri quadrilateral has the following properties: (a) Its diagonals are congruent.

(b) Its summit angles are congruent and acute.

(c) Its midsegment is perpendicular to both the base and the summit.

(d) It is a parallelogram.

(e) It is a convex quadrilateral.

4.         Every Lambert quadrilateral has the following properties: (a) Its fourth angle is acute.

(b) It is a parallelogram.

(c) It is a convex quadrilateral.

5.         Define the defect 6 of a triangle 」ABC by

6(」ABC) = 180 − (A + B + C).

If 」ABC is a triangle, D is a point on BC, 6(」ABD) = 61 and 6(」ACD) = 62 , what is 6(」ABC)?

6.         Use (3) to prove the AAA Theorem: Two triangles whose corresponding angles   are equal are congruent. This is what might be called an Angle-Angle-Angle Theorem.   No such result is true in Euclidean geometry. It can also be stated as saying that there are no similar triangles in hyperbolic geometry (unless they are congruent), since triangles    with the same angles must be the same size.

7.          Find the area in the upper-half plane  H of the super triangle with vertices i, 1, and

1 + √2.

8.         Use one of the hyperbolic models to define parallel and super parallel lines. How do we find the minimum distance between two super parallel lines? Explain.


2022-12-09