TOC & Ch. 0 & Ch. 1 Axiom Table of Contents Ch. 0 Introduction Ch. 1 Axiomatic Systems 1.1.1 Introduction 1.1.2 Examples 1.1.3 History 1.2 A Finite Geometry 1.3 Finite Projective 1.4 Applications

Ch. 2 Neutral Geometry Ch. 2 Table of Contents 2.1.1 Introduction 2.1.2 History 2.1.3 Analytic Models 2.2 Incidence Axioms 2.3 Distance/Ruler Axioms 2.4.1 Plane Separation Axiom 2.4.2 Angle & Measure 2.5.1 Supplement Postulate 2.5.2 SAS Postulate 2.6.1 Parallel Lines 2.6.2 Saccheri Quadrilateral 2.7.1 Euclid Parallel Postulate 2.7.2 Hyperbolic Parallel Postulate 2.7.3 Elliptic Parallel Postulate 2.8 Euclid/Hyperbolic/Elliptic Birkhoff's Axioms Hilbert's Axioms SMSG Axioms

Ch. 3 Transformational Ch. 3 Table of Contents 3.1.1 Introduction 3.1.2 History 3.2.1 Definitions 3.2.2 Analytic Model 3.2.3 Affine Transformation 3.3.1 Isometry 3.3.2 Model/Collinearity 3.3.3 Model/Isometry 3.4.1 Direct Isometry 3.4.2 Model/Direct 3.5.1 Indirect Isometry 3.5.2 Model/Indirect 3.6.1 Similarity Transformation 3.6.2 Model/Similarity 3.7 Other Affine Transformations

Ch. 4 Projective Geometry Ch. 4 Table of Contents 4.1.1 Introduction 4.1.2 Historical 4.2.1 Axioms 4.2.2 Basic Theorems 4.3 Duality 4.4 Desargue's Theorem 4.5.1 Harmonic Sets 4.5.2 Music & Harmonic Sets 4.6.1 Definitions for Projectivity 4.6.2 Fundamental Theorem 4.6.3 Projectivity/Harmonic Sets 4.6.4 Alternate Construction 4.7.1 Conics 4.7.2 Pascal's Theorem 4.7.3 Tangents to Conics

Other Topics Ch. 5 Spherical Geometry Ch. 6 Fractal Geometry Ch. 7 Topology

Appendices Internet Resources Index Geometer's Sketchpad/GeoGebra JavaSketchpad/GeoGebraHTML Video Lectures Logic Review References Acknowledgements

              2.2 Incidence Axioms   Printout
Geometry enlightens the intellect and sets one's mind right.
—Ibn Khaldun (1332–1406)

 

      Why do surveyors, photographers, and artists use tripods? How is the location of a football that has been punted out of bounds determined?  The axioms for Euclidean geometry are such that the mathematics matches our real-world expectations. What axioms or theorems justify the solutions to the two questions?
      The SMSG incidence axioms are Postulates 1 and 5–8; however, since we are only concerned with plane geometry, the only axioms that apply to our study of a neutral geometry are Postulates 1, 5(a), and 6.

 

Postulate 1. (Line Uniqueness) Given any two distinct points there is exactly one line that contains them.

 

Postulate 5(a). (Existence of Points) Every plane contains at least three noncollinear points.

Postulate 6. (Points on a Line Lie in a Plane) If two points lie in a plane, then the line containing these points lies in the same plane.

Definition. A set of points S is collinear if there is a line l such that S is a subset of l.  S is noncollinear if S is not a collinear set. If {A, B, C} is a collinear set, we say that the points A, B, and C are collinear.

Theorem 2.2. Two distinct lines intersect in at most one point.
     

Proposition 2.3. The Cartesian plane satisfies SMSG Postulates 1, 5(a), and 6.

Proof. To show Postulate 5(a) is satisfied consider the three points (0, 0), (1, 0), and (0, 1), which are points in the Cartesian plane. The three points are not on the same vertical line, since they do not have the same first coordinate. Suppose they are on a nonvertical line l_m,b, then 0 = m(0) + b, 0 = m(1) + b, and 1 = m(0) + b, i.e. b = 1 and b = 0, which is a contradiction. Hence (0, 0), (1, 0), and (0, 1) are three distinct noncollinear points in the Cartesian plane, and SMSG Postulate 5 is satisfied.   
      Next, show SMSG Postulate 1 is satisfied. Let P(x₁, y₁) and Q(x₂, y₂) be distinct points in the Cartesian plane. Then x₁ and x₂ are either equal or not equal. We first show the existence of a line containing P and Q.
Case 1. Assume x₁ = x₂. Set a = x₁ = x₂. Thus P and Q are both on line .
Case 2. Assume x₁ and x₂ are not equal. We need to find m and b such that P and Q are on line l_m,b.  (First, do some scratch work to find an m and b that will work. Based on the scratch work, define an m and b, and then show they work.) Set  and  We show that P and Q are on the line . For P,

 

Hence P is on line l_m,b. For Q,  . Hence Q is on line l_m,b. Therefore, P and Q are on line l_m,b. Thus by the two cases, there is at least one line that contains P and Q.
      We need to show there is exactly one line that contains P and Q. Suppose P and Q belong to lines l and k. There are three possible cases (1) both lines are vertical; (2) both lines are nonvertical; or (3) one line is vertical, and the other line is nonvertical.
Case 1. Assume l = l_a and k = k_b. Then a = x₁ = x₂ = b. Hence l and k are the same line.
Case 2. Assume l = l_m,b and k = k_n,c. Then y₁ = mx₁ +b, y₂ = mx₂ +b, y₁ = nx₁ +c, and y₂ = nx₂ +c. Solve this system of four equations:

mx₁ +b = nx₁ +c and mx₂ +b = nx₂ +c

(m – n) x₁ = c – b and (m – n) x₂ = c – b

(m – n)x₁ = (m – n)x₂

(m – n)(x₁ – x₂) = 0.

Since for nonvertical lines x₁ and x₂ are not equal, we have that m = n. Thus c – b = 0, i.e. c = b. Hence l = l_m,b = k_n,c = k.
Case 3. Without loss of generality, assume l = l_a and k = k_n,c. Since P and Q are on l, a = x₁ = x₂. That is, P = (a, y₁) and Q = (a, y₂). Since P and Q are on k, y₁ = nx₁ + c = na + c = nx₂ + c = y₂. Hence P = (x₁, y₁) = (x₂, y₂) = Q. But this contradicts that P and Q are distinct. Therefore, this case is not possible.
      By the above three cases, there is exactly one line that contains P and Q. Since the two points were arbitrarily chosen, we have that the Cartesian plane satisfies SMSG Postulate 1.
      From how lines are defined for a Cartesian plane and the above proof that Postulate 1 is satisfied, it immediately follows that the Cartesian plane satisfies SMSG Postulate 6.// 

 

Corollary to Proposition 2.3. The Euclidean plane, Taxicab plane, and Max-distance plane satisfy SMSG Postulates 1, 5(a), and 6.

 

Exercise 2.11. Find the axioms from a high school geometry book that correspond to SMSG Postulates 1 and 5.

Exercise 2.12. Prove Theorem 2.2.

Exercise 2.13. Show the Missing Strip Plane satisfies (a) SMSG Postulate 1 and (b) SMSG Postulate 5(a).

 

Exercise 2.14. Show the Poincaré Half-plane satisfies (a) SMSG Postulate 1 and (b) SMSG Postulate 5(a).

Exercise 2.15. Does a Discrete plane satisfy (a) SMSG Postulate 1?  Justify. (b) SMSG Postulate 5(a)? Justify.

Exercise 2.16. Do the Riemann Sphere and Modified Riemann Sphere satisfy (a) SMSG Postulate 1? Justify. (b) SMSG Postulate 5(a)? Justify.