Euclidean and Non Euclidean Geometry

Euclid’s Postulates on the Sphere

Which of Euclid’s Five Postulates are true on the Sphere? Also look at the following postulate:

Postulate 1’. Two points determine a unique line.

For each postulate, either provide an explanation of why the postulate would hold or a counter example showing that the postulate is false. I strongly suggest using pictures to supplement your explanations.

Ans:

Here are the postulates:

1. A (unique) straight line may be drawn from any point to any other point.

Postulate 1’. Two points determine a unique line.

Although a line can be drawn in between any two points in sphere, it cannot be unique. We can prove this by drawing two different straight lines through two points. E.g., many longitudes passing through North and South Pole. Therefore, this postulate is not true in a sphere.

2. Every limited straight line can be extended indefinitely to a (unique) straight line.

A spherical geometry does not follow this postulate. This can be proven as a straight line (great circles) ends where it starts and could not be extended indefinitely.

3. A circle may be drawn with any center and any distance.

This is false. As using any center and distance longer than the radius of the sphere, a circle cannot be drawn. As it can be seen in the drawing, radius longer than the radius of sphere produces no circle.

4. All right angles are equal.

Right angle is angle formed between two perpendicular lines. As perpendicular lines exist on a sphere, we can say that right angles are equal in a sphere. This could be proven by coinciding two right angles together.

5. If a straight line intersecting two straight lines makes the interior angles on the same side less than two right angles, then the two lines (if extended indefinitely) will meet on that side on which the angles are less than two right angles.

As, the lines meets on both directions, irrespective of sum of interior angles, this is not true on a sphere. As you can see below, the straight lines meet at both sides.

3.1 What Is an Angle?

Give some possible definitions of the term “angle.” Do all these definitions apply to the plane as well as to spheres'? What are the advantages and disadvantages of each? For each definition, what does it mean for two angles to be congruent? How can we check?

Ans:

Angles can be defined in three perspectives.

Dynamic: The change of direction caused between two lines by turning, rotating, etc., could be considered as an angle. This definition will help define the change of direction precisely but fails to measure when the lines are not changing direction. We could check if two angles are congruent if a line rotated by these angles produces the same final lines.

Measure: Angle could be the length of an arc or ratio between areas of circular sections. This definition could give us a way of giving values to an angle but fails to define when the lines are changing direction. Verifying that two different measures have the same value, the congruency of two angles could be checked.

Geometric shape: Angle could be the geometry created by the intersection of two lines. This could be checked by using isometries. If two angles could coincide with each other, they are congruent.

3.2 Vertical Angle Theorem (VAT)

Prove: Opposite angles formed by two intersecting straight lines are congruent. [Note: Angles such as a and b are called vertical. What properties of straight lines and/or the plane are you using in your proof? Does your proof also work on a sphere? Why? Which definitions from Problem 3.2 are you using in your proof?

Ans:

Consider a straight-line AB. If we were to rotate the line AB from the midpoint of AB, say O, clockwise with a certain angle α, the image produced will be A’B’ line. Here the angle between AOA’(α) is the same as the angle formed between BOB’ (β) as it was created by the same action of rotating the line AB. Hence, α= β. This method should work in a sphere as rotation could be performed in a sphere. Here the angle could be is defined dynamically, as the change of direction caused between two lines. We define straight lines symmetrically. A straight line is defined to have the line into itself if it was to be reflected from the axis perpendicular to it.