Answer the following questions. Many of the answers have clues by the diagrams given in the problem but you should confirm the findings by drawing your own figures on the Lenart sphere (starting with question 5).
Definition of a sphere
1) Spherical geometry is geometry on a sphere. Define a sphere.
Note: In spherical geometry you will be working on the surface of the sphere and not in the interior of the sphere
Shortest Path
2) Locate two points in the plane and label them \(P\) and \(Q\). What is the shortest path between two points on a plane?
Definition of a sphere
1) Spherical geometry is geometry on a sphere. Define a sphere.
Note: In spherical geometry you will be working on the surface of the sphere and not in the interior of the sphere
Shortest Path
2) Locate two points in the plane and label them \(P\) and \(Q\). What is the shortest path between two points on a plane?
3) Locate two points on the sphere and label them \(A\) and \(B\). (Do not locate these points such that they are opposite each other on the sphere. Such opposite points are called antipodal points and they will be referred to later on in the activity.)
What is the shortest distance between two points on a sphere?
What is the shortest distance between two points on a sphere?
Segments
4) If \(\overline{PQ}\) (from problem #2 in the plane were extended indefinitely beyond the graph, how far would it go?
5) If you extended \(\overline{AB}\) on your sphere, describe what the result would be.
6) You have just drawn a Great Circle. Define a Great Circle and name a Great Circle on Earth.
7) Latitudes are the horizontals on the earth that determine north from south, (see picture above). Are all lines of latitude Great Circles?
8) Longitudes are verticals on the earth that connect the North Pole with the South Pole and that determine east from west, (see picture above). Are all lines of longitudes Great Circles?
9) In the Euclidean plane the shortest path from \(P\) to \(Q\) is unique, and its measure is fixed. Can the same be said of the \(\overline{AB}\) on the sphere? Is the measure of a \(\overline{AB}\) on a sphere unique?
Measuring segments
To measure the distance between two points \(P\) and \( Q\) in the Euclidean plane, you would use a ruler or perhaps the distance formula. The units of measurement would be a linear standard unit of measurement (i.e. inches, centimeters, miles, etc.).
In spherical geometry the distance between two points is measured in degrees, that is a fraction of the Great Circle which contains the segment that connects the two points.
In this problem we will discuss how distances are measured on a sphere. Suppose the Earth is a sphere. In Euclidean space the Earth has a radius of \(6,400\) km (the radius in this case as measured from the center of the sphere to any point on the surface of Earth is \(6,400\) km).
10) What is Earth’s circumference?
11) How many degrees does this represent?
12) If two places on Earth are opposite each other (i.e. poles or antipodal points), what is the distance between them in kilometers? In degrees?
13) If two places are \(90^{\circ}\) apart from each other, how far apart are they in kilometers?
14) If two places are \(5026\) km apart, what is their distance apart measured in degrees (nearest degree?
15) Mars has a circumference of \(21,320\) kilometers. What does this distance represent in degrees?
16) What is the furthest distance that two places on Mars can be apart from each other in degrees? In kilometers (in the spherical sense)?
In the following problems, use the Lenart sphere to investigate your results
Euclid’s First Postulate
17) Euclid’s first postulate states that for every point \(P\) and every point \(Q\), where \(P\) is not equal to \(Q\), there exists a unique line \(\ell\) through \(P\) and \(Q\). On the Lenart sphere, draw two points and connect them with the shortest path possible. Is Euclid’s first postulate valid in spherical geometry?
5) If you extended \(\overline{AB}\) on your sphere, describe what the result would be.
6) You have just drawn a Great Circle. Define a Great Circle and name a Great Circle on Earth.
7) Latitudes are the horizontals on the earth that determine north from south, (see picture above). Are all lines of latitude Great Circles?
8) Longitudes are verticals on the earth that connect the North Pole with the South Pole and that determine east from west, (see picture above). Are all lines of longitudes Great Circles?
9) In the Euclidean plane the shortest path from \(P\) to \(Q\) is unique, and its measure is fixed. Can the same be said of the \(\overline{AB}\) on the sphere? Is the measure of a \(\overline{AB}\) on a sphere unique?
Measuring segments
To measure the distance between two points \(P\) and \( Q\) in the Euclidean plane, you would use a ruler or perhaps the distance formula. The units of measurement would be a linear standard unit of measurement (i.e. inches, centimeters, miles, etc.).
In spherical geometry the distance between two points is measured in degrees, that is a fraction of the Great Circle which contains the segment that connects the two points.
In this problem we will discuss how distances are measured on a sphere. Suppose the Earth is a sphere. In Euclidean space the Earth has a radius of \(6,400\) km (the radius in this case as measured from the center of the sphere to any point on the surface of Earth is \(6,400\) km).
10) What is Earth’s circumference?
11) How many degrees does this represent?
12) If two places on Earth are opposite each other (i.e. poles or antipodal points), what is the distance between them in kilometers? In degrees?
13) If two places are \(90^{\circ}\) apart from each other, how far apart are they in kilometers?
14) If two places are \(5026\) km apart, what is their distance apart measured in degrees (nearest degree?
15) Mars has a circumference of \(21,320\) kilometers. What does this distance represent in degrees?
16) What is the furthest distance that two places on Mars can be apart from each other in degrees? In kilometers (in the spherical sense)?
In the following problems, use the Lenart sphere to investigate your results
Euclid’s First Postulate
17) Euclid’s first postulate states that for every point \(P\) and every point \(Q\), where \(P\) is not equal to \(Q\), there exists a unique line \(\ell\) through \(P\) and \(Q\). On the Lenart sphere, draw two points and connect them with the shortest path possible. Is Euclid’s first postulate valid in spherical geometry?
Intersecting Points of Two Lines
18) In how many points can two lines on a plane intersect?
18) In how many points can two lines on a plane intersect?
19) Use your sphere to draw two Great Circles on the sphere. In how many points can two lines on the sphere intersect?
Remember, in spherical geometry Great Circle = Line
Euclid’s Fifth Postulate (Parallel Postulate)
20) Euclid’s fifth postulate state: “Given a line \(\ell\) and a point \(P\) not on \(\ell\), there exists a unique line though \(P\) parallel to \(\ell\).” We know that the definition of parallelism is lines that do not intersect. Do parallel lines exist in spherical geometry?
How would you re-word Euclid’s Fifth Postulate so that it is true for spherical geometry?
Betweenness
21) Locate a point \(R\) between two points \(P\) and \(Q\) on the plane.
Euclid’s Fifth Postulate (Parallel Postulate)
20) Euclid’s fifth postulate state: “Given a line \(\ell\) and a point \(P\) not on \(\ell\), there exists a unique line though \(P\) parallel to \(\ell\).” We know that the definition of parallelism is lines that do not intersect. Do parallel lines exist in spherical geometry?
How would you re-word Euclid’s Fifth Postulate so that it is true for spherical geometry?
Betweenness
21) Locate a point \(R\) between two points \(P\) and \(Q\) on the plane.
The Betweenness Axiom states that if \(P, Q,\) and \(R\) are three points in the plane, then one and only one point is between the other two. Draw a Great Circle on your sphere and locate a point \(C\) between points \(A\) and \(B\) on the sphere.
Is the Betweenness Axiom valid for the three points that are drawn on the sphere?
Is the Betweenness Axiom valid for the three points that are drawn on the sphere?
Euclid’s Second Postulate
22) Euclid’s second postulate states: “a line segment can be extended infinitely from each side.” Is this postulate valid in spherical geometry? Please explain.
Circle on a sphere
23) A circle is defined as the set of points equidistant from a given point
22) Euclid’s second postulate states: “a line segment can be extended infinitely from each side.” Is this postulate valid in spherical geometry? Please explain.
Circle on a sphere
23) A circle is defined as the set of points equidistant from a given point
Draw a circle on your sphere using your compass set at \(45^{\circ}\). Label the two centers of the circle (yes, there are two centers). The first radii is \(45^{\circ}\). Measure the second radii. How are the radii lengths related to each other?
Euclid’s Third Postulate
24) Euclid’s third postulate states, “A circle can be drawn with any center and any radius. Is this true for circles on the sphere?
Vertical Angles
25) We know that when two lines intersect on the plane, the measures of the vertical angles are congruent. We can confirm that using a protractor.
Euclid’s Third Postulate
24) Euclid’s third postulate states, “A circle can be drawn with any center and any radius. Is this true for circles on the sphere?
Vertical Angles
25) We know that when two lines intersect on the plane, the measures of the vertical angles are congruent. We can confirm that using a protractor.
Draw two Great Circles on your sphere. Label the points of intersection \(A\) and \(B\). Use your spherical protractor to measure the pairs of vertical angles formed at the point of intersection of the Great Circles. What do you notice about each pair of vertical angles?
Perpendiculars
Draw a Euclidean line. Locate a point \(P\) that is not on the line. What is the shortest path from the point to the line? This path is called the distance from \(P\) to the line. Construct this path.
Perpendiculars
Draw a Euclidean line. Locate a point \(P\) that is not on the line. What is the shortest path from the point to the line? This path is called the distance from \(P\) to the line. Construct this path.
26) Draw Great Circle on the sphere. Locate a point \(P\) that is not on the Great Circle and not a pole point. What is the distance (shortest path) from the point to the arc? How many perpendiculars can be drawn from the point \(P\) to the arc?
Locate a pole point for this Great Circle. How many perpendiculars can be drawn from this pole to the Great Circle?
27) Draw two intersecting lines l and m on the plane. Can you draw a common perpendicular to these two lines? Draw two parallel lines l and m on the plane. Can you draw a common perpendicular to these two lines? If you can, how many can be drawn? 28) Draw two great circles that are not perpendicular. Can a common perpendicular be drawn to these two Great Circles? |
Intersecting points of Three lines
29) In how many different ways can three lines intersect in the plane?
30) In how many points do two geodesics lines (Great Circles) on a sphere intersect (Figure 1)? In how many ways do three geodesics lines (Great Circles) on the sphere intersect (Figure 2 & Figure 3)? Note, you may want to construct these circle on your sphere.
29) In how many different ways can three lines intersect in the plane?
30) In how many points do two geodesics lines (Great Circles) on a sphere intersect (Figure 1)? In how many ways do three geodesics lines (Great Circles) on the sphere intersect (Figure 2 & Figure 3)? Note, you may want to construct these circle on your sphere.
Answer Bank