Download Spherical geometry

SOME ELEMENTARY CONCEPTS ABOUT
SPHERICAL GEOMETRY
• The simplest shape in spherical geometry is the spherical point, just as the
planar point is the simplest shape in plane geometry. Each spherical point
determines another point on the sphere called the opposite point.
Example: On a globe, the North and South poles are a spherical point and
its opposite point.
• The simplest line is the great circle, the largest of all the circles that can be
drawn on the sphere. The great circle on the sphere is the spherical equivalent
of the straight line on the plane.
Example: On a globe, the equator and the globes longitudes are great
circles. The globe’s latitudes are NOT.
• Two different great circles always intersect in two opposite points of
intersection. Parallel great circles do not exist. You can draw only one great
circle through two spherical points that are not opposite. If the points are
opposite, then there are infinitely many great circles passing through them.
• Two points on a great circle divide it into two separate and measurable arcs. If
the two points are not opposite, then we define their spherical distance as the
measure of the shorter of the two arcs. We use degrees to measure distance
along a great circle. If the two points are opposite, then their spherical distance
measures 180°.
• Each great circle has two centers on the sphere, called the pole points or poles
of the great circle. As a pole, each spherical point determines one and only one
great circle, called the equator.
Lenart, Istvan: Non-Euclidean Adventures on the Lenart Sphere – 1995
SOME COMPARISONS BETWEEN THE PLANE AND THE SPHERE
ON THE PLANE
ON THE SPHERE
The straight line is infinite.
The great circle is finite.
The straight line has no center.
The great circle has 2 centers
There is one unique straight line that passes
through any pair of points.
There is one unique great circle passing through
any pair of points unless the points are pole
points.
A line segment is the shortest path between two
points.
An arc of a great circle is the shortest path
between two points.
When measuring distance between two points
along a straight line, there is only one distance
you can measure.
When measuring distance between two points
along the great circle, there are 2 distances you
can measure. We usually measure the shorter
one.
Two distinct lines with no point of intersection are Two distinct circles can never be parallel.
called parallel lines.
Two distinct lines have at most one point of
intersection.
Two distinct great circles have exactly two points
of intersection.
A pair of perpendicular lines intersects one times
and creates four right angles.
A pair of perpendicular great circles intersects two
times and creates eight right angles.
A pair of parallel straight lines have one common
perpendicular(s).
A pair of parallel great circles does not exist.
A two-sided polygon (does not) exist.
A two-sided polygon (does ) exist.
Three noncollinear points determine one unique
triangle.
Three noncollinear points (that are not opposite)
determine eight unique triangles.
The angle sum of a triangle is exactly 180
degrees.
The interior angle sum of a triangle ranges from
180 to 540 degrees
Lesson 1: The Two-Sided Polygon
Learning Objectives:
1.
Students will be able to define a polygon on a sphere.
2.
Students will be able to construct a two-sided polygon on a sphere.
3.
Students will explore the characteristics of a two-sided polygon.
Materials:
Classroom set of Lenart Spheres
Worksheet – Lab 1 (attached)
Procedures:
1. Ask the students to define a planar polygon. To help them, display
several figures and have them determine which ones are polygons, which aren’t and why.
2.
Have students formulate (by exploration, if necessary) a definition for a spherical polygon.
3.
Explain why a two-sided polygon does not exist in plane geometry.
4.
Have students try to create a two-sided polygon on a sphere, sketch a picture and describe
what it looks like. Then ask if they can determine a name for this figure, based on their
knowledge of plane polygons (triangle – 3 sides, quadrilateral – 4 sides, BIANGLE – 2 sides).
5.
Split the students in to groups of two or three. Using the Lenart sphere, complete the
attached lab worksheet.
Note: The students should notice that, in a spherical biangle, the two angles will always be
congruent, and the angle measure of each angle will not exceed 180 degrees.
Note: Students should have prior knowledge of and experience with the Lenart Sphere
ANSWER KEY FOR:
SPHERICAL GEOMETRY -- LAB 1
EXPLORATION OF TWO-SIDED SPHERICAL POLYGONS
A. Explore, using the Lenart sphere, different angle values and angle sums for a two-sided polygon. Record the
values in the table.
Angle 1
Angle 2
Angle Sum
ANSWERS WILL VARY, BUT ANGLE 1 WILL ALWAYS EQUAL ANGLE 2. ANGLE SUM MUST NOT
EXCEED 360 DEGREES.
B. Construct a two-sided polygon having each of the following conditions, if possible. Sketch the polygon and
describe its characteristics. Or, if not possible, explain why. Remember, you must always take the minor arcs.
ANSWERS FOR CHARACTERISTICS MAY VARY
1) Two 45° angles
Sketch:
2) Two 90° angles
Sketch:
Characteristics or why it’s not possible:
Characteristics or why it’s not possible:
The two arcs will intersect at 45° angles.
The two arcs will intersect at 90° angles.
3) Exactly one 90° angle
Sketch:
4) At least one 200° angle
Sketch:
NOT POSSIBLE
NOT POSSIBLE
Characteristics or why it’s not possible:
Characteristics or why it’s not possible:
The two angles will always be congruent.
It is not possible to create only one 90
degree angle
You must always take the minor arc.
An angle will never exceed 180 degrees.
C. Make a generalization about what you noticed. What types of angles could not be created? What did you
notice about the angle sums of the biangles? Use complete sentences.
ANSWERS MAY VARY
The two angles will always be congruent. Each angle will never exceed 180 degrees. Therefore, the angle
sum will never exceed 360 degrees.
D. A ray on a plane corresponds to an arc on a sphere. For each of the following, circle the correct answer.
In a plane
1. Two distinct rays will (always, sometimes, never) meet.
On a Sphere
1. Two arcs will (always, sometimes, never) meet.
2. A two-sided polygon (does, does not) exist.
2. A two-sided polygon (does, does not) exist.
3. The angle sum of a triangle is (equal to,
more than, less than) 180°.
3. The angle sum of a biangle is always (180°,
more than 180°, less than or equal to 360°).
SPHERICAL GEOMETRY -- LAB 1
EXPLORATION OF TWO-SIDED SPHERICAL POLYGONS
A. Explore, using the Lenart sphere, different angle values and angle sums for a two-sided polygon. Record the
values in the table.
Angle 1
Angle 2
Angle Sum
B. Construct a two-sided polygon having each of the following conditions, if possible. Sketch the polygon and
describe its characteristics. Or, if not possible, explain why. Remember, you must always take the minor arcs.
1) Two 45° angles
Sketch:
2) Two 90° angles
Sketch:
Charactistics or why it’s not possible:
Charactistics or why it’s not possible:
______________________________
______________________________
______________________________
______________________________
______________________________
______________________________
4) Exactly one 90° angle
Sketch:
4) At least one 200° angle
Sketch:
Charactistics or why it’s not possible:
Charactistics or why it’s not possible:
______________________________
______________________________
______________________________
______________________________
______________________________
______________________________
C. Make a generalization about what you noticed. What types of angles could not be created? What did you notice
about the angle sums of the biangles? Use complete sentences.
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
D. A ray on a plane corresponds to an arc on a sphere. For each of the following, circle the correct answer.
In a plane
On a Sphere
1. Two distinct rays will (always, sometimes, never) meet.
1. Two arcs will (always, sometimes, never) meet.
2. A two-sided polygon (does, does not) exist.
2. A two-sided polygon (does, does not) exist.
3. The angle sum of a triangle is (equal to,
more than, less than) 180°.
3. The angle sum of a biangle is always (180°,
more than 180°, less than or equal to 360°).
Lesson 2a: Exploration of Triangles on a Sphere
Learning Objectives:
4.
Students will explore various angle sums of spherical triangles.
5.
Students will make conjectures about angle sums of spherical triangles.
Materials:
Ruler or straightedge
Paper
Protractor
Classroom set of Lenart Spheres
Worksheet – Lab 2a (attached)
Procedures:
6.
Have students draw construct three different triangles on their paper using their straightedge.
7.
Using their protractors, have students measure and label each angle, then find the angle sum
of each triangle.
8.
Discuss why the angle sum is always 180 degrees.
9.
Split the students in to groups of two or three. Using the Lenart sphere, complete the
attached lab worksheet.
Note: The students should recall from Lab 1 that angle sums of biangles can be 360 degrees, so
the angle sums of triangles should be even greater.
SPHERICAL GEOMETRY -- LAB 2a
EXPLORATION OF SPHERICAL TRIANGLES
B.
Explore, using the Lenart sphere, the angle values and angle sums of seven different
triangles. Use a variety of angle values. Record the values in the table .
Triangle 1
Triangle 2
Triangle 3
Triangle 4
Triangle 5 Triangle 6
Triangle 7
Angle 1
Angle 2
Angle 3
Angle Sum
C. Use your observations to answer the following questions.
1. What do you think is the smallest possible angle sum for a spherical triangle, and why?
Discuss the location of the vertices.
________________________________________________________________________
________________________________________________________________________
2. What is the largest possible angle sum, and why? Discuss the location of the vertices.
________________________________________________________________________
________________________________________________________________________
3. Make a conjecture: What do you think is the largest possible angle sum for a quadrilateral
on a sphere? The smallest? Explain why you believe this.
________________________________________________________________________
________________________________________________________________________
4. Test your conjecture using the Lenart sphere. Has your belief changed or been
confirmed? Explain.
________________________________________________________________________
________________________________________________________________________
ANSWER KEY FOR:
SPHERICAL GEOMETRY -- LAB 2a
EXPLORATION OF SPHERICAL TRIANGLES
A. Explore, using the Lenart sphere, the angle values and angle sums of seven different
triangles. Use a variety of angle values. Record the values in the table .
Triangle 1
Triangle 2
Triangle 3
Triangle 4
Triangle 5 Triangle 6
Triangle 7
Angle 1
Angle 2
Angle 3
Angle Sum
ANSWERS ABOVE WILL VARY. ANGLE SUMS WILL RANGE FROM 180 TO 540 DEGREES. EACH
ANGLE MEASURE MUST NOT EXCEED 180 DEGREES.
B. Use your observations to answer the following questions.
1. What do you think is the smallest possible angle sum for a spherical triangle, and why?
Discuss the location of the vertices.
SMALLEST: 180 DEGREES, WHEN ALL 3 VERTICES LIE ON THE SAME GREAT CIRCLE.
ONE SIDE LIES ATOP THE OTHER TWO, LEAVING TWO 0 DEGREE ANGLES AND 1 180
DEGREE ANGLE.
2. What is the largest possible angle sum, and why? Discuss the location of the vertices.
LARGEST: 540 DEGREES, WHEN ALL THREE VERTICES LIE ON THE SAME GREAT
CIRCLE, AND EACH ANGLE MEASURES 180 DEGREES.
3. Make a conjecture: What do you think is the largest possible angle sum for a quadrilateral
on a sphere? The smallest? Explain why you believe this.
_______________________ANSWERS WILL VARY______________________________
________________________________________________________________________
4. Test your conjecture using the Lenart sphere. Has your belief changed or been
confirmed? Explain.
ANSWERS WILL VARY. STUDENTS SHOULD DISCOVER THAT ANGLE SUMS RANGE
FROM 360 TO 720 DEGREES.
Lesson 2b: Exploration of Right Triangles on a Sphere
Learning Objectives:
6.
Students will explore right triangles and the possibility of having more than one right angle.
7.
Students will make comparisons about plane right triangles and spherical right triangles.
Materials:
Ruler or straightedge
Paper
Protractor
Classroom set of Lenart Spheres
Worksheet – Lab 2b (attached)
Procedures:
10. Discuss what a right triangle is on the plane.
11. Ask students if it is possible for a plane triangle to have more than one right angle. They
should recall from the previous lab that the angle sum of a plane triangle is180 degrees.
12. Using their protractor and straightedge, have them construct a figure with two right angles
and explain why this will never be a triangle.
13. Split the students in to groups of two or three. Using the Lenart sphere, complete the
attached lab worksheet.
SPHERICAL GEOMETRY -- LAB 2b
EXPLORATION OF SPHERICAL RIGHT TRIANGLES
D.
Using the Lenart sphere, construct four different right triangles. Two triangles should have
two right angles, and two triangles should have just one right angle. Record the values in the
table .
Triangle 1
Triangle 2
Triangle 3
Triangle 4
Angle 1
Angle 2
Angle 3
Angle Sum
Length of
Side 1
Length of
Side 2
Length of
Side 3
E. Construct an equilateral triangle, whose angles are each 90 degrees. Determine the angle
sum of this triangle and also the lengths of its sides.
Angle sum:________________________
Side 2:____________________________
Side 1:____________________________
Side 3:____________________________
F. Does the Pythagorean Theorem hold for spherical triangles? Use the data you recorded
above to investigate, then explain your reasoning.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
ANSWER KEY FOR:
SPHERICAL GEOMETRY -- LAB 2b
EXPLORATION OF SPHERICAL RIGHT TRIANGLES
A. Using the Lenart sphere, construct four different right triangles. Two triangles should have
two right angles, and two triangles should have just one right angle. Record the values in the
table.
ANSWERS WILL VARY. ANGLE SUMS WILL RANGE FROM 180 TO 540
DEGREES.
Triangle 1
Triangle 2
Triangle 3
Triangle
4
Angle 1
Angle 2
Angle 3
Angle Sum
Length of
Side 1
Length of
Side 2
Length of
Side 3
B. Construct an equilateral triangle, whose angles are each 90 degrees. Determine the angle
sum of this triangle and also the lengths of its sides.
Angle sum: 270 DEGREES
Side 2: 90 DEGREES_______________
Side 1: 90 DEGREES_______________
Side 3: 90 DEGREES_______________
C. Does the Pythagorean Theorem hold for spherical triangles? Use the data you recorded
above to investigate, then explain your reasoning.
BASED ON THE DATA IN PART B, THE PYTHAGOREAN THEOREM DOES NOT
HOLD TRUE, BECAUSE ALL THREE SIDES ARE THE SAME LENGTH.
Lesson 3: Assessment of Spherical Geometry
Learning Objectives:
8.
Students will retrieve information they learned about spherical geometry from Lab 1, Lab2a,
and Lab 2b and additional classroom exercises.
2. Students will compare and contrast characteristics of a plane and a sphere.
Materials:
Classroom set of Lenart Spheres
Worksheet – Lab 3 (attached)
Procedures:
14. Have a class discussion about what was learned during the previous labs.
15. Have students comment on their likes and dislikes of spherical geometry.
16. Split the students in to groups of 2 or 3. Have them complete the worksheet – Lab 3.
SPHERICAL GEOMETRY -- LAB 3
ASSESSMENT LAB
Using the knowledge you gained from the previous labs and other lessons taught during class, fill
in the chart below, comparing characteristics on a plane to those on a sphere.
ON THE PLANE
ON THE SPHERE
The straight line is infinite.
The great circle is ________________
The straight line has no center.
The great circle has ____________________
There is one unique straight line that passes
through any pair of points.
There is one unique great circle passing through
any pair of points unless
_________________________________
A/an __________________________ is the
shortest path between two points.
A/an _________________________ is the
shortest path between two points.
When measuring distance between two points
along a straight line, there is only one distance
you can measure.
When measuring distance between two points
along the great circle, there are _______
distances you can measure. We usually measure
the _________________ one.
Two distinct lines with no point of intersection are Two distinct circles
called _____________________________
_________________________________
Two distinct lines have at most ________ point(s) Two distinct great circles have _____________
of intersection.
point(s) of intersection.
A pair of perpendicular lines intersects _____
time(s) and creates ______ right angles.
A pair of perpendicular great circles intersects
______ times and creates _____ right angles.
A pair of parallel straight lines have ___________ A pair of parallel great
common perpendicular(s).
circles__________________________________
___________________________
A two-sided polygon (does
does not) exist.
A two-sided polygon (does
does not) exist.
Three noncollinear points determine ______
unique triangle(s).
Three noncollinear points (that are not opposite)
determine _________ unique triangle(s).
The angle sum of a triangle is ____________.
The interior angle sum of a triangle ranges from
_________ to ___________.
ANSWER KEY FOR:
SPHERICAL GEOMETRY -- LAB 3
ASSESSMENT LAB
ON THE PLANE
ON THE SPHERE
The straight line is infinite.
The great circle is finite.
The straight line has no center.
The great circle has 2 centers
There is one unique straight line that passes
through any pair of points.
There is one unique great circle passing through
any pair of points unless the points are pole
points.
A/an line segment is the shortest path between
two points.
A/an arc of a great circle is the shortest path
between two points.
When measuring distance between two points
along a straight line, there is only one distance
you can measure.
When measuring distance between two points
along the great circle, there are 2 distances you
can measure. We usually measure the shorter
one.
Two distinct lines with no point of intersection are Two distinct circles can never be parallel.
called parallel lines.
Two distinct lines have at most one point(s) of
intersection.
Two distinct great circles have exactly two
point(s) of intersection.
A pair of perpendicular lines intersects one
time(s) and creates four right angles.
A pair of perpendicular great circles intersects
two times and creates eight right angles.
A pair of parallel straight lines have one common A pair of parallel great circles does not exist.
perpendicular(s).
A two-sided polygon (does
does not) exist.
A two-sided polygon (does
does not) exist.
Three noncollinear points determine one unique
triangle(s).
Three noncollinear points (that are not opposite)
determine eight unique triangle(s).
The angle sum of a triangle is exactly 180
degrees.
The interior angle sum of a triangle ranges from
180 to 540 degrees