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Section 3.2 Distance and the Ruler Postuláte 39 Corollary 3.2.8. A * C * B if and only if B * C * A. Proof. Let A, B, and C be three points such that C E AB (hypothesis). If A * C * B, then AC + CB = AB (definition). Since AB ВА, АС CA, and CB = BC, it is also the case that BC + CA = BA. Therefore, B * C * A. The proof of the converse is similar. Any function that has the properties spelled out in Theorem 3.2.7 can be used to measure distances. Such a function is called a metric. Definition 3.2.9. A metric is a function D :P X P → R such that 1. D(P, Q) = D(Q, P) for every P and Q, 2. D( P, Q) > 0 for every P and Q, and 3. D(P, Q) = 0 if and only if P Q. %3D A version of the triangle inequality D(P, Q) < D(P, R) + D(R, Q) is often included as part of the definition of metric, but we do not include it in the definition because we will prove the triangle inequality as a theorem in the next chapter. The familiar distance formula from calculus and high school geometry is an example of a metric. EXAMPLE 3.2.10 The Euclidean metric Define the distance between points (x1, yı) and (x2, y2) in the Cartesian plane by d((x1, yı). (x2, y2)) = V(x2 - x1)2 + (y2 - yı)2. | This metric is called the Euclidean metric The verification that d satisfies the conditions

Section 3.2 Distance and the Ruler Postuláte 39 Corollary 3.2.8. A * C * B if and only if B * C * A. Proof. Let A, B, and C be three points such that C E AB (hypothesis). If A * C * B, then AC + CB = AB (definition). Since AB ВА, АС CA, and CB = BC, it is also the case that BC + CA = BA. Therefore, B * C * A. The proof of the converse is similar. Any function that has the properties spelled out in Theorem 3.2.7 can be used to measure distances. Such a function is called a metric. Definition 3.2.9. A metric is a function D :P X P → R such that 1. D(P, Q) = D(Q, P) for every P and Q, 2. D( P, Q) > 0 for every P and Q, and 3. D(P, Q) = 0 if and only if P Q. %3D A version of the triangle inequality D(P, Q) < D(P, R) + D(R, Q) is often included as part of the definition of metric, but we do not include it in the definition because we will prove the triangle inequality as a theorem in the next chapter. The familiar distance formula from calculus and high school geometry is an example of a metric. EXAMPLE 3.2.10 The Euclidean metric Define the distance between points (x1, yı) and (x2, y2) in the Cartesian plane by d((x1, yı). (x2, y2)) = V(x2 - x1)2 + (y2 - yı)2. | This metric is called the Euclidean metric The verification that d satisfies the conditions

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Section 3.2 Distance and the Ruler Postulate 39 Corollary 3.2.8. A * C * Bif and only if B * C * A. Proof. Let A, B, and C be three points such that C E AB (hypothesis). If A * C * B, then AC + CB = AB (definition). Since AB the case that BC + CA = BA. Therefore, B * C * A. The proof of the converse is similar. ВА, АС СА, and CB BC, it is also Any function that has the properties spelled out in Theorem 3.2.7 can be used to measure distances. Such a function is called a metric. Definition 3.2.9. A metric is a function D : P × P → R such that 1. D(P, Q) 2. D(P, Q) > 0 for every P and Q, and 3. D( P, Q) = 0 if and only if P = Q. D(Q, P) for every P and Q, A version of the triangle inequality D( P, Q) < D(P, R) + D(R, Q) is often included as part of the definition of metric, but we do not include it in the definition because we will prove the triangle inequality as a theorem in the next chapter. The familiar distance formula from calculus and high school geometry is an example of a metric. EXAMPLE 3.2.10 The Euclidean metric Define the distance between points (x1, yı) and (x2, y2) in the Cartesian plane by d((x1, yı), (x2, y2)) = J(x2 x1)2 + (y2 y1 )2. This metric is called the Euclidean metric. The verification that d satisfies the conditions in the definition of metric is left as an exercise (Exercise 2). There are other ways in which to measure distances in the Cartesian plane. Here is an example that will be important later in the chapter. EXAMPLE 3.2.11 The taxicab metric Define the distance between points (x1, y1) and (x2, y2) in the Cartesian plane by p((x1, y1), (x2, y2)) = |x2 x1l + ly2 - yı|. This metric is called the taxicab metric. The verification that p satisfies the conditions in the definition of metric is left as an exercise (Exercise 3). The name taxicab is given to the second metric because the distance from (x1, y1) to (x2, y2) is measured by traveling along a line parallel to the x-axis and then along one parallel to the y-axis in much the same way that rectangular grid of streets and cannot travel directly from point A to point B. There is also a natural way to measure distances between points on the sphere. taxicab in Manhattan must follow a What we are calling a metric should more properly be called a semimetric. However, all the examples of metrics we will consider satisfy the triangle inequality and are metrics according to the standard definition. We omit the triangle inequality from the definition only because it is an unnecessary assumption in this context. 40 Chapter 3 Axioms for Plane Geometry (X, )
Transcribed Image Text:Section 3.2 Distance and the Ruler Postulate 39 Corollary 3.2.8. A * C * Bif and only if B * C * A. Proof. Let A, B, and C be three points such that C E AB (hypothesis). If A * C * B, then AC + CB = AB (definition). Since AB the case that BC + CA = BA. Therefore, B * C * A. The proof of the converse is similar. ВА, АС СА, and CB BC, it is also Any function that has the properties spelled out in Theorem 3.2.7 can be used to measure distances. Such a function is called a metric. Definition 3.2.9. A metric is a function D : P × P → R such that 1. D(P, Q) 2. D(P, Q) > 0 for every P and Q, and 3. D( P, Q) = 0 if and only if P = Q. D(Q, P) for every P and Q, A version of the triangle inequality D( P, Q) < D(P, R) + D(R, Q) is often included as part of the definition of metric, but we do not include it in the definition because we will prove the triangle inequality as a theorem in the next chapter. The familiar distance formula from calculus and high school geometry is an example of a metric. EXAMPLE 3.2.10 The Euclidean metric Define the distance between points (x1, yı) and (x2, y2) in the Cartesian plane by d((x1, yı), (x2, y2)) = J(x2 x1)2 + (y2 y1 )2. This metric is called the Euclidean metric. The verification that d satisfies the conditions in the definition of metric is left as an exercise (Exercise 2). There are other ways in which to measure distances in the Cartesian plane. Here is an example that will be important later in the chapter. EXAMPLE 3.2.11 The taxicab metric Define the distance between points (x1, y1) and (x2, y2) in the Cartesian plane by p((x1, y1), (x2, y2)) = |x2 x1l + ly2 - yı|. This metric is called the taxicab metric. The verification that p satisfies the conditions in the definition of metric is left as an exercise (Exercise 3). The name taxicab is given to the second metric because the distance from (x1, y1) to (x2, y2) is measured by traveling along a line parallel to the x-axis and then along one parallel to the y-axis in much the same way that rectangular grid of streets and cannot travel directly from point A to point B. There is also a natural way to measure distances between points on the sphere. taxicab in Manhattan must follow a What we are calling a metric should more properly be called a semimetric. However, all the examples of metrics we will consider satisfy the triangle inequality and are metrics according to the standard definition. We omit the triangle inequality from the definition only because it is an unnecessary assumption in this context. 40 Chapter 3 Axioms for Plane Geometry (X, )
10:57 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB FIGURE 3.7: Two circles in the rational plane coordinate functions use real numbers will allow us to fill that gap. Specifically, we will prove that if one circle contains a point that is inside a second circle and another point that is outside the second circle, then the two circles must intersect. Even using the powerful axioms in this chapter the proof is not easy, however, and we will not attempt that particular proof until Chapter 8. XERCISES 3.2 1. Prove: If l and m are two lines, the number of points in l n m is either 0, 1, or o. 2. Show that the Euclidean metric defined in Example 3.2.10 is a metric (i.e., verify that the function d satisfies the three conditions in the definition of metric on page 39). 3. Show that the taxicab metric defined in Example 3.2.11 is a metric (i.e., verify that the function p satisfies the three conditions in the definition of metric on page 39). 4. Show that the spherical metric defined in Example 3.2.12 is a metric (i.e., verify that the function s satisfies the three conditions in the definition of metric on page 39). 5. Betweenness in taxicab geometry. Let A = (0, 0), В %3D (1,0) and C %3D (1, 1) and let p denote the taxicab metric. (a) Find all points P such that p(A, P) + p(P, B) = p(A, B). Draw a sketch in the Cartesian plane. (b) Find all points P such that p( A, P) + p( P, C) = p(A, C). Draw a sketch in the Cartesian plane. 6. Betweenness on the sphere. Let A, B, and C be points on the sphere S². Define C to be between A and B if A, B, and C are collinear and s(A, C) + s(C, B) = s(A, B), where s is the metric defined in Example 3.2.12. Also define segment in the usual way. (a) Find all points that are between A and C in case A and C are nonantipodal points. Sketch the segment from A to C. (b) Find all points that are between A and C in case A and C are antipodal points. Sketch the segment from A to C. 7. Find all points (x, y) in R² such that p( (0, 0), (x, y)) = 1, where p is the taxicab metric. Draw a sketch in the Cartesian plane. (This shape might be called a “circle" in the taxicab metric.) 8. The square metric. Define the distance between two points (x1, y1) and (x2, y2) in R² by D((x1, yı ), (x2, y2)) = max{|x2 (a) Verify that D is a metric. (b) Find all points (x, y) in R² such that D((0,0), (x, y)) Cartesian plane. (This should explain the name square metric.) x1), ly2 yı|}.4 = 1. Draw a sketch in the 4max{a, b} denotes the larger of the two real numbers a and b. 46 Chapter 3 Axioms for Plane Geometry 9. Verify that the functions defined in Example 3.2.14 are coordinate functions. 10. Verify that the functions defined in Example 3.2.15 are coordinate functions. 11. Let l be a line in the Cartesian plane R2. Find a function f :l → R that is a coordinate function for l in the square metric. (See Exercise 8 for definition of square metric.) 12. Assume that f :l → R is a coordinate function for le. (a) Prove that – f is also a coordinate function for l. (b) Prove that g : l → R defined by g( P) = f(P) + c for some constant c is also a coordinate function for l. (c) Prove that if h : l → R is any coordinate function for l then there must exist a constant c such that either h(P) = f(P) + c or h( P) = -f(P) + c. 13. Let l be a line and let f : l → R be a function such that PQ = \f(P) every P, Q E l. Prove that f is a coordinate function for l. 14. Prove the following fact from high school algebra that was needed in the Theorem 3.2.17: If x and y are two nonzero real numbers such that |x| + |y| then either both x and y are positive or both x and y are negative. 15. Prove Corollary 3.2.20. 16. Prove that if C E AB and C # A, then AB = 17. Prove existence and uniqueness of midpoints (Theorem 3.2.22). AĆ.
Transcribed Image Text:10:57 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB FIGURE 3.7: Two circles in the rational plane coordinate functions use real numbers will allow us to fill that gap. Specifically, we will prove that if one circle contains a point that is inside a second circle and another point that is outside the second circle, then the two circles must intersect. Even using the powerful axioms in this chapter the proof is not easy, however, and we will not attempt that particular proof until Chapter 8. XERCISES 3.2 1. Prove: If l and m are two lines, the number of points in l n m is either 0, 1, or o. 2. Show that the Euclidean metric defined in Example 3.2.10 is a metric (i.e., verify that the function d satisfies the three conditions in the definition of metric on page 39). 3. Show that the taxicab metric defined in Example 3.2.11 is a metric (i.e., verify that the function p satisfies the three conditions in the definition of metric on page 39). 4. Show that the spherical metric defined in Example 3.2.12 is a metric (i.e., verify that the function s satisfies the three conditions in the definition of metric on page 39). 5. Betweenness in taxicab geometry. Let A = (0, 0), В %3D (1,0) and C %3D (1, 1) and let p denote the taxicab metric. (a) Find all points P such that p(A, P) + p(P, B) = p(A, B). Draw a sketch in the Cartesian plane. (b) Find all points P such that p( A, P) + p( P, C) = p(A, C). Draw a sketch in the Cartesian plane. 6. Betweenness on the sphere. Let A, B, and C be points on the sphere S². Define C to be between A and B if A, B, and C are collinear and s(A, C) + s(C, B) = s(A, B), where s is the metric defined in Example 3.2.12. Also define segment in the usual way. (a) Find all points that are between A and C in case A and C are nonantipodal points. Sketch the segment from A to C. (b) Find all points that are between A and C in case A and C are antipodal points. Sketch the segment from A to C. 7. Find all points (x, y) in R² such that p( (0, 0), (x, y)) = 1, where p is the taxicab metric. Draw a sketch in the Cartesian plane. (This shape might be called a “circle" in the taxicab metric.) 8. The square metric. Define the distance between two points (x1, y1) and (x2, y2) in R² by D((x1, yı ), (x2, y2)) = max{|x2 (a) Verify that D is a metric. (b) Find all points (x, y) in R² such that D((0,0), (x, y)) Cartesian plane. (This should explain the name square metric.) x1), ly2 yı|}.4 = 1. Draw a sketch in the 4max{a, b} denotes the larger of the two real numbers a and b. 46 Chapter 3 Axioms for Plane Geometry 9. Verify that the functions defined in Example 3.2.14 are coordinate functions. 10. Verify that the functions defined in Example 3.2.15 are coordinate functions. 11. Let l be a line in the Cartesian plane R2. Find a function f :l → R that is a coordinate function for l in the square metric. (See Exercise 8 for definition of square metric.) 12. Assume that f :l → R is a coordinate function for le. (a) Prove that – f is also a coordinate function for l. (b) Prove that g : l → R defined by g( P) = f(P) + c for some constant c is also a coordinate function for l. (c) Prove that if h : l → R is any coordinate function for l then there must exist a constant c such that either h(P) = f(P) + c or h( P) = -f(P) + c. 13. Let l be a line and let f : l → R be a function such that PQ = \f(P) every P, Q E l. Prove that f is a coordinate function for l. 14. Prove the following fact from high school algebra that was needed in the Theorem 3.2.17: If x and y are two nonzero real numbers such that |x| + |y| then either both x and y are positive or both x and y are negative. 15. Prove Corollary 3.2.20. 16. Prove that if C E AB and C # A, then AB = 17. Prove existence and uniqueness of midpoints (Theorem 3.2.22). AĆ.
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