# Notes on spherical geometry

Save this PDF as:

Size: px
Start display at page:

Download "Notes on spherical geometry"

## Transcription

1 Notes on spherical geometry Math 130 Course web site: This handout covers some spherical trigonometry (following yan s exposition) and the topic of dual triangles. 1. Some tools Trig identities We will need to remember some trigonometric identities. In particular, we should recall cos(a + b) cos(a) cos(b) sin(a) sin(b) (1) cos(a b) cos(a) cos(b) + sin(a) sin(b) and the related formula ( X + Y cos(x ) cos(y ) 2 sin 2 ) ( ) X Y sin. (2) 2 Cross products For vectors A and B in 3, the cross-product A B can be characterized geometrically by the conditions: A B is zero if A and B are linearly dependent; otherwise: A B is orthogonal to both A and B;

2 A, B and A B form a right-handed basis; the length of A B is A B sin θ, where θ is the angle between the vectors. Algebraically, if then The inner product a 1 is the determinant of the matrix b 1 A a 2 and B b 2 a 3 b 3 a 2 b 3 a 3 b 2 A B a 3 b 1 a 1 b 3. a 1 b 2 a 2 b 1 A, B C a 1 b 1 c 1 a 2 b 2 c 2. a 3 b 3 c 3 From the familiar symmetries of the determinant, we can deduce A, B C B,C A. (3) The vector cross-product is not associative, so (A B) C and A (B C ) are usually different. There is a very useful formula involving these triple products: (A B) C A,C B B,C A. (4) 2. Spherical geometry Incidence geometry on the sphere We write S 2 for the unit sphere in 3 : S 2 { x 3 x 1 }.

3 By a line in spherical geometry, we will mean a great circle on the sphere S 2. In spherical geometry, any two distinct lines meet in exactly two (antipodal) points. If P and Q are points in S 2 that are neither equal nor antipodal, then there is a unique line containing both of them. To set down some formulae for these things, we first note that we can described a great circle in S 2 as the set of unit vectors that are orthogonal to a given unit vector ξ in 3. That is, every line l in spherical geometry can be written as l { x S 2 x, ξ 0 }, for some ξ in S 2. The point ξ is called a pole for the line l. The line l is called the polar line for the point ξ. For each line l, there are two possible choices for the pole: if ξ is one, then ξ is the other. To compute the line that goes through two given points on S 2, we proceed as follows. If P and Q are in S 2 and are neither equal nor parallel, then they are linearly independent vectors in 3 ; and P Q is therefore non-zero. The point ξ P Q P Q is therefore a point on the unit sphere which is orthogonal to the vectors P and Q. It is the pole corresponding to the unique line through P and Q. To compute the intersection of two distinct lines l and m in S 2, we take ξ and η to be poles for the two lines; then the points of l m are the points on S 2 that are orthogonal to both ξ and η. Again, the cross-product allows us to construct these points: the two points of intersection of l and m are ± ξ η ξ η. Distances and angles on the sphere Let P and Q be distinct points on the sphere S 2. If P and Q are not antipodal, then there is exactly one line (great circle) passing through both points. The points themselves divide the great circle into two segments. The shorter of these is called the minor segment. The distance from P to Q is the length of this minor segment. If

4 P and Q are antipodal, then all great-circle segments from P to Q have length π, and in this case d(p,q ) is π. All this can be captured more simply by the formula d(p,q ) cos 1 P,Q. Now let P, Q and R are three points on S 2. Suppose that Q ±P and Q ±R. This means that there are unique minor segments joining Q to P and Q to R. How should we understand the angle between these to segments, where they meet at the point Q? The great circles through QP and QR are the intersection of the sphere with two planes in 3, and the angle we seek is the angle between these two planes. This is the same as the angle between vectors orthogonal to these two planes. So we must measure the angle between the poles corresponding to these great circles. To get the internal angle θ PQR at Q, we must make the correct choice for the poles. The right choice is to take the poles for these two lines to be Q P Q P and Q R Q R.

5 This leads to the formula for the angle, Q P θ cos 1 Q P, Q R. (5) Q R As a check of the signs, observe that when P R, the inner product on the right is 1 (not 1), so that the angle θ comes out as 0, which is what we want. (The wrong choice of poles might have led to the inner product being and the angle being π; this would have been the external angle at Q, instead of the internal angle.) Spherical triangles By a spherical triangle we mean a triple A, B, C of non-collinear points on S 2 together with the minor segments joining them. (The fact that A, B and C are not collinear implies in particular that no one can be the antipode of the another.) We write the side-lengths of the triangle as a d(b,c ) b d(c, A) c d(a, B).

6 We write the angles of the triangle as α C AB β ABC γ BC A. The cosine rule The cosine rule in Euclidean geometry expresses a relation between four quantities associated to a Euclidean triangle: the lengths of the three sides, and one of the angles. There is a version of the cosine rule in spherical geometry. Let ABC be as above. We shall express the cos(α) in terms of the lengths of the sides. We start with (5), which gives cos α A B A B, A C. A C Next we use the fact that A B is sin of the angle between the unit vectors A and B, which is the side-length c of the spherical triangle. Treating A C similarly, we get 1 cos α A B, A C. Now we use (3) and (4), obtaining 1 cos α A, B (A C ) 1 A, (C A) B 1 A, C, B A A, B C 1 ( ) B,C A, A A, B A,C 1 ( ) B,C A,C A, B Finally we remember that A,C is cos of the spherical distance d(a,c ), etcetera, and we end up with: cos a cosb cosc cos α. (6)

7 This is the spherical cosine rule. We can rewrite it as: cos a cosb cosc + cos α. Let us see what happens when the side-lengths of the triangle are all small. In this case, we approximate cos a by 1 a 2 /2 and sinb by b. We get or more simply, 1 2 a2 /2 1 2 (b2 + c 2 ) + cos αbc, a 2 b 2 + c 2 2ab cos α. We recognize this as the cosine rule from Euclidean geometry: the formulae from spherical geometry coincide with the Euclidean formulae to leading order when the triangle is small. The sine rule Having obtained a formula for cos α, we can obtain a formula now for sin α. We start with sin α (1 cos α)(1 + cos α) M N. We then calculate (using the cosine rule (6) above), M 1 cos α ( + cos a cosb cos a )/ () ( + cosb cosc cos a )/ (). Next we apply our cosine formulae (1) and (2) to get M + cosb cosc cos a cos(b c) cos a sinb ( sinc ) ( ) 2 sin b c+a 2 sin b c a 2 2 sin(s c) sin(s b).

8 where s (a + b + c)/2. Similarly, cosb cosc + cos a N cos(b + c) + cos a sinb ( sinc) ( ) 2 sin a+b+c 2 sin a b c 2 2 sin(s) sin(s a). Putting it all together and diving by sin 2 a, we get sin 2 α sin 2 a 4sin(s) sin(s a) sin(s b) sin(s c) (sin a ) 2. The key point now is that the expression on the right is symmetric in a, b and c. We therefore deduce: sin α sin a sin β sinb sin γ sinc. (7) This is the spherical sine rule. As for the cosine rule, when a, b and c are all small, this formula coincides to leading order with the Euclidean sine rule: sin α a sin β b sin γ c. 3. Duality The dual triangle Let ABC again be a spherical triangle. To each of the vertices A, B and C of our spherical triangle, we can associate its polar line: three great circles l A, l B and l C. And to each side of the triangle, BC, C A and AB, we can associate a point: the pole for the corresponding great circle (though we have a choice of two). In order to nail down the choices a little, we proceed as follows. We will suppose that the vertices of our triangle ABC are labelled so that the unit vectors A, B, C

9 in 3 form a right-handed basis. This is equivalent to asking that As pole for the line AB, we take the point A B,C > 0. (8) A B A B. (This is one of the two possible poles, but we choose this one.) For the other two lines BC and C A, we use the same recipe, permuting the letters cyclically. Thus the three chosen poles for the three sides of the triangle are the points ξ a B C B C ξ b C A C A ξ c A B A B. The dual triangle to the triangle ABC is the triangle ξ a ξ b ξ c whose vertices are these poles corresponding to the sides of the original triangle. Thus, to every spherical triangle, we have assigned a dual triangle. The dual of the dual The relationship between a triangle and its dual is a symmetrical one. That is, if we form the dual of the triangle ξ a ξ b ξ c, we will end up with the original triangle ABC again. We can verify this as follows. If we form the dual of the triangle ξ a ξ b ξ c, we will get a triangle A B C where (for example) C ξ a ξ b ξ a ξ b. What we must show is that C C. We observe that C is a positive multiple of ξ a ξ b, which in turn is a positive multiple of (B C ) (C A).

10 We use the identity (4) to write this last vector as B,C A C C,C A B B,C A C 0 A B,C C The coefficient A B,C is positive, because of our condition that A, B, C is a right-handed basis (8). Putting it all together, we see that C is a positive multiple of C; and since both are unit vectors, it follows that C C. Thus the dual of the dual triangle is the original triangle. Left or right handed? In showing that the dual of the dual is the original triangle, we used the fact that (8) was a positive quantity: the right-handed condition. What would have happened if the original triple A, B, C had been a left-handed basis? It turns out that ξ a, ξ b, ξ c (the vertices of the dual triangle) will be a right-handed basis of 3 no matter what. When we take the dual of the dual, we will end up with A, B and C that are again right-handed: these points will be the antipodal points to the original triangle A, B, C. Side-lengths and angles of the dual triangle Let us compute the side-lenghts of the dual triangle. The distance between the vertices ξ b, ξ c of the dual triangle is given by the usual formula for spherical distance: d(ξ b, ξ c ) cos 1 ξ b, ξ c. From the formulae defining ξ b and ξ c, we see that this becomes C A d(ξ b, ξ c ) cos 1 C A, A B A B ) A C cos ( 1 A C, A B A B A C π cos 1 A C, A B A B π BAC π α.

11 from (5). Thus the side-lengths of the dual triangle are π α, π β, π γ, where α, β and γ are the angles of the original triangle. What about the angles of the dual triangle? We could do another calculation; or we could exploit the fact that the dual of the dual is the original. That is, we reverse the role of the original triangle and its dual in the above statement, and we see that the side-lengths of the original triangle are π θ, π φ, π ψ, where θ, φ and ψ are the angles of the dual triangle. estating this: the angles of the dual triangle are π a, π b, π c, where a, b and c are the side-lengths of the original triangle. The dual cosine rule We can apply the cosine rule (6) to the dual triangle ξ a ξ b ξ c. The role of α is now played by the angle of the dual triangle at the vertex ξ a, which is π a, and so on. Thus we obtain cos(π a) cos(π α) cos(π β) cos(π γ ), sin(π β) sin(π γ ) which simplifies to cos a cos α + cos β cos γ sin β sin γ, (9) This is the dual cosine rule. It expresses the side-lengths of the triangle ABC (e.g. the side-length a) in terms of the angles of the triangle. Note that it is a special feature of spherical geometry (not shared with Euclidean geometry) that the side-lengths of a triangle are determined by the angles. An example Consider a regular pentagon on the sphere. Specifically, let us think of five points P 1,..., P 5 in the northern hemisphere, all the same distance from the north pole; these are the vertices of our pentagon. The edges are arcs of great circles, and all five edges have the same length. Suppose that the internal angles of the pentagon are all 120 (that is, 2π/3). How long are the edges?

12 Let N be the north pole and let us look at the triangle P 1 P 2 N. The line from N to P 1 bisects the internal angle of the pentagon at P 1. So N P 1 P 2 π/3. Similarly P 1 P 2 N π/3. At N the rays from all the P i meet at equal angles, so P 1 N P 2 2π/5. If x d(p 1, P 2 ), then from the dual cosine rule above we get cos x cos(2π/5) + cos2 (π/3) sin 2 (π/3) cos(2π/5) + 1/4 3/4 4 cos(2π/5) So x cos 1 ( 5/3), which is about 0.73.

### VECTOR NAME OF THE CHAPTER PART-C FIVE MARKS QUESTIONS PART-E TWO OR FOUR QUESTIONS PART-A ONE MARKS QUESTIONS

NAME OF THE CHAPTER VECTOR PART-A ONE MARKS QUESTIONS PART-B TWO MARKS QUESTIONS PART-C FIVE MARKS QUESTIONS PART-D SIX OR FOUR MARKS QUESTIONS PART-E TWO OR FOUR QUESTIONS 1 1 1 1 1 16 TOTAL MARKS ALLOTED

More information

### HYPERBOLIC TRIGONOMETRY

HYPERBOLIC TRIGONOMETRY STANISLAV JABUKA Consider a hyperbolic triangle ABC with angle measures A) = α, B) = β and C) = γ. The purpose of this note is to develop formulae that allow for a computation of

More information

### Higher Geometry Problems

Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

More information

### Modern Geometry Homework.

Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z

More information

### Converting wind data from rotated lat-lon grid

Converting wind data from rotated lat-lon grid Carsten Hansen, Farvandsvæsenet Copenhagen, 16 march 2001 1 Figure 1: A Hirlam Grid Hirlam output data are in a rotated lat-lon grid The Hirlam grid is typically

More information

### Spherical coordinates 1

Spherical coordinates 1 Spherical coordinates Both the earth s surface and the celestial sphere have long been modeled as perfect spheres. In fact, neither is really a sphere! The earth is close to spherical,

More information

### VECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a.

VECTOR ALGEBRA Chapter 10 101 Overview 1011 A quantity that has magnitude as well as direction is called a vector 101 The unit vector in the direction of a a is given y a and is represented y a 101 Position

More information

### Pre-Calculus Review Problems Solutions

MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry Pre-Calculus Review Problems Solutions Problem 1. Give equations for the following lines in both point-slope and slope-intercept form. (a) The

More information

### M243. Fall Homework 2. Solutions.

M43. Fall 011. Homework. s. H.1 Given a cube ABCDA 1 B 1 C 1 D 1, with sides AA 1, BB 1, CC 1 and DD 1 being parallel (can think of them as vertical ). (i) Find the angle between diagonal AC 1 of a cube

More information

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

More information

### Application of Trigonometry in Engineering 1

Name: Application of Trigonometry in Engineering 1 1.1 Laboratory (Homework) Objective The objective of this laboratory is to learn basic trigonometric functions, conversion from rectangular to polar form,

More information

### Week 12 The Law of Cosines and Using Vectors

Week 12 The Law of Cosines and Using Vectors Overview This week we look at a law that handles some of the situations that the Law of Sines could not. The Law of Cosines can be used as well, in part, to

More information

### MATH 304 Linear Algebra Lecture 24: Scalar product.

MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent

More information

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

More information

### Laboratory 2 Application of Trigonometry in Engineering

Name: Grade: /26 Section Number: Laboratory 2 Application of Trigonometry in Engineering 2.1 Laboratory Objective The objective of this laboratory is to learn basic trigonometric functions, conversion

More information

### Announcements. 2-D Vector Addition

Announcements 2-D Vector Addition Today s Objectives Understand the difference between scalars and vectors Resolve a 2-D vector into components Perform vector operations Class Activities Applications Scalar

More information

### Problem Set 1 Solutions Math 109

Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twenty-four symmetries if reflections and products of reflections are allowed. Identify a symmetry which is

More information

### THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA

THE LAWS OF COSINES FOR NON-EUCLIDEAN TETRAHEDRA B.D.S. MCCONNELL Darko Veljan s article The 500-Year-Old Pythagorean Theorem 1 discusses the history and lore of probably the only nontrivial theorem in

More information

### Constructions with Compass and Straightedge

MODULE 6 Constructions with Compass and Straightedge A thing constructed can only be loved after it is constructed: but a thing created is loved before it exists. Gilbert Keith Chesterton 1. Constructible

More information

### 13.4 THE CROSS PRODUCT

710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

More information

### BASIC GEOMETRY GLOSSARY

BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that

More information

### MTH 122 Plane Trigonometry Fall 2015 Test 1

MTH 122 Plane Trigonometry Fall 2015 Test 1 Name Write your solutions in a clear and precise manner. Answer all questions. 1. (10 pts) a). Convert 44 19 32 to degrees and round to 4 decimal places. b).

More information

### 2. Equilateral Triangles

2. Equilateral Triangles Recall the well-known theorem of van Schooten. Theorem 1 If ABC is an equilateral triangle and M is a point on the arc BC of C(ABC) then MA = MB + MC. Proof Use Ptolemy on the

More information

### 6. Angles. a = AB and b = AC is called the angle BAC.

6. Angles Two rays a and b are called coterminal if they have the same endpoint. If this common endpoint is A, then there must be points B and C such that a = AB and b = AC. The union of the two coterminal

More information

### *** START OF THIS PROJECT GUTENBERG EBOOK SPHERICAL TRIGONOMETRY ***

The Project Gutenberg ebook of Spherical Trigonometry, by I. Todhunter This ebook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away

More information

### Applications of Trigonometry

chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced

More information

### Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

More information

### Isosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors

Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A

More information

### Are right spherical triangles wrong?

Are right spherical triangles wrong? Emily B. Dryden Bucknell University Bowdoin College April 13, 2010 Basic objects Unit sphere: set of points that are 1 unit from the origin in R 3 Straight lines great

More information

### 5. Orthic Triangle. Remarks There are several cyclic quadrilaterals

5. Orthic Triangle. Let ABC be a triangle with altitudes AA, BB and CC. The altitudes are concurrent and meet at the orthocentre H (Figure 1). The triangle formed by the feet of the altitudes, A B C is

More information

### REGULAR POLYGONS. Răzvan Gelca Texas Tech University

REGULAR POLYGONS Răzvan Gelca Texas Tech University Definition. A regular polygon is a polygon in which all sides are equal and all angles are equal. Definition. A regular polygon is a polygon in which

More information

### Rotation Matrices. Suppose that 2 R. We let

Suppose that R. We let Rotation Matrices R : R! R be the function defined as follows: Any vector in the plane can be written in polar coordinates as rcos, sin where r 0and R. For any such vector, we define

More information

### a a. θ = cos 1 a b ) b For non-zero vectors a and b, then the component of b along a is given as comp

Textbook Assignment 4 Your Name: LAST NAME, FIRST NAME (YOUR STUDENT ID: XXXX) Your Instructors Name: Prof. FIRST NAME LAST NAME YOUR SECTION: MATH 0300 XX Due Date: NAME OF DAY, MONTH DAY, YEAR. SECTION

More information

### There exists at most one parallel to a given line through a given point. Two lines can but need not have some points in common.

Math 3181 Name: Dr. Franz Rothe February 6, 2014 All3181\3181_spr14t1.tex Test has to be turned in this handout. 1 Solution of Test 10 Problem 1.1. As far as two-dimensional geometry is concerned, Hilbert

More information

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

More information

### SAT Subject Math Level 2 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

More information

### Paths Between Points on Earth: Great Circles, Geodesics, and Useful Projections

Paths Between Points on Earth: Great Circles, Geodesics, and Useful Projections I. Historical and Common Navigation Methods There are an infinite number of paths between two points on the earth. For navigation

More information

### THREE DIMENSIONAL GEOMETRY

Chapter 11 THREE DIMENSIONAL GEOMETRY 111 Overview 1111 Direction cosines of a line are the cosines of the angles made by the line with positive directions of the co-ordinate axes 111 If l, m, n are the

More information

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

More information

### 11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

More information

### 4.1 Euclidean Parallelism, Existence of Rectangles

Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles Definition 4.1 Two distinct

More information

### PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin

PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles

More information

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

### 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

More information

### Foundations of Geometry 1: Points, Lines, Segments, Angles

Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

More information

### Logic and Incidence Geometry

Logic and Incidence Geometry February 27, 2013 1 Informal Logic Logic Rule 0. No unstated assumption may be used in a proof. 2 Theorems and Proofs If [hypothesis] then [conclusion]. LOGIC RULE 1. The following

More information

### If a question asks you to find all or list all and you think there are none, write None.

If a question asks you to find all or list all and you think there are none, write None 1 Simplify 1/( 1 3 1 4 ) 2 The price of an item increases by 10% and then by another 10% What is the overall price

More information

### Notes from February 11

Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The

More information

### I V.5 : Theorems of Desargues and Pappus

I V.5 : Theorems of Desargues and Pappus The elegance of the[se] statements testifies to the unifying power of projective geometry. The elegance of the[ir] proofs testifies to the power of the method of

More information

### High School Math Contest

High School Math Contest University of South Carolina January 31st, 015 Problem 1. The figure below depicts a rectangle divided into two perfect squares and a smaller rectangle. If the dimensions of this

More information

### Geometry Unit 1. Basics of Geometry

Geometry Unit 1 Basics of Geometry Using inductive reasoning - Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture- an unproven statement that is based

More information

### Axiom A.1. Lines, planes and space are sets of points. Space contains all points.

73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach

More information

### Chapter 1. Foundations of Geometry: Points, Lines, and Planes

Chapter 1 Foundations of Geometry: Points, Lines, and Planes Objectives(Goals) Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in

More information

### 3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled?

1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, co-ordinate geometry (which connects algebra

More information

### Problems and Solutions, INMO-2011

Problems and Solutions, INMO-011 1. Let,, be points on the sides,, respectively of a triangle such that and. Prove that is equilateral. Solution 1: c ka kc b kb a Let ;. Note that +, and hence. Similarly,

More information

### Chapter 1: Essentials of Geometry

Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

More information

### 1. Introduction identity algbriac factoring identities

1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as

More information

### Betweenness of Points

Math 444/445 Geometry for Teachers Summer 2008 Supplement : Rays, ngles, and etweenness This handout is meant to be read in place of Sections 5.6 5.7 in Venema s text [V]. You should read these pages after

More information

### ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Trigonometry

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH0000 SEMESTER 1 016/017 DR. ANTHONY BROWN 5. Trigonometry 5.1. Parity and co-function identities. In Section 4.6 of Chapter 4 we looked

More information

### DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

More information

### P222. Optics Supplementary Note # 3: Mirrors Alex R. Dzierba Indiana University. θ θ. Plane Mirrors

P222 Optics Supplementary Note # 3: Mirrors Alex R. Dzierba Indiana University Plane Mirrors Let s talk about mirrors. We start with the relatively simple case of plane mirrors. Suppose we have a source

More information

### Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

More information

### CARTESIAN VECTORS AND THEIR ADDITION & SUBTRACTION

CARTESIAN VECTORS AND THEIR ADDITION & SUBTRACTION Today s Objectives: Students will be able to: a) Represent a 3-D vector in a Cartesian coordinate system. b) Find the magnitude and coordinate angles

More information

### 5 Hyperbolic Triangle Geometry

5 Hyperbolic Triangle Geometry 5.1 Hyperbolic trigonometry Figure 5.1: The trigonometry of the right triangle. Theorem 5.1 (The hyperbolic right triangle). The sides and angles of any hyperbolic right

More information

### Section 8 Inverse Trigonometric Functions

Section 8 Inverse Trigonometric Functions Inverse Sine Function Recall that for every function y = f (x), one may de ne its INVERSE FUNCTION y = f 1 (x) as the unique solution of x = f (y). In other words,

More information

### 6.7. The sine and cosine functions

35 6.7. The sine and cosine functions Surprisingly enough, angles and other notions of trigonometry play a significant role in the study of some biological processes. Here we review some basic facts from

More information

### Math 311 Test III, Spring 2013 (with solutions)

Math 311 Test III, Spring 2013 (with solutions) Dr Holmes April 25, 2013 It is extremely likely that there are mistakes in the solutions given! Please call them to my attention if you find them. This exam

More information

### Pythagorean theorems in the alpha plane

MATHEMATICAL COMMUNICATIONS 211 Math. Commun., Vol. 14, No. 2, pp. 211-221 (2009) Pythagorean theorems in the alpha plane Harun Baris Çolakoğlu 1,, Özcan Gelişgen1 and Rüstem Kaya 1 1 Department of Mathematics

More information

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

More information

### Geometry A Solutions. Written by Ante Qu

Geometry A Solutions Written by Ante Qu 1. [3] Three circles, with radii of 1, 1, and, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to

More information

### Plane transformations and isometries

Plane transformations and isometries We have observed that Euclid developed the notion of congruence in the plane by moving one figure on top of the other so that corresponding parts coincided. This notion

More information

### Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

More information

### Orthogonal Matrices. u v = u v cos(θ) T (u) + T (v) = T (u + v). It s even easier to. If u and v are nonzero vectors then

Part 2. 1 Part 2. Orthogonal Matrices If u and v are nonzero vectors then u v = u v cos(θ) is 0 if and only if cos(θ) = 0, i.e., θ = 90. Hence, we say that two vectors u and v are perpendicular or orthogonal

More information

### Definition 12 An alternating bilinear form on a vector space V is a map B : V V F such that

4 Exterior algebra 4.1 Lines and 2-vectors The time has come now to develop some new linear algebra in order to handle the space of lines in a projective space P (V ). In the projective plane we have seen

More information

### Mathematics. Unit 4: Trigonometric Identities

Georgia Standards of Excellence Curriculum Frameworks GSE Pre-Calculus Mathematics Unit 4: Trigonometric Identities These materials are for nonprofit educational purposes only. Any other use may constitute

More information

### Straight Line motion with rigid sets

Straight ine motion with rigid sets arxiv:40.4743v [math.mg] 9 Jan 04 Robert Connelly and uis Montejano January, 04 Abstract If one is given a rigid triangle in the plane or space, we show that the only

More information

### Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.

More information

### 9.3. Direction Ratios and Direction Cosines. Introduction. Prerequisites. Learning Outcomes. Learning Style

Direction Ratios and Direction Cosines 9.3 Introduction Direction ratios provide a convenient way of specifying the direction of a line in three dimensional space. Direction cosines are the cosines of

More information

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

### Assignment 3. Solutions. Problems. February 22.

Assignment. Solutions. Problems. February.. Find a vector of magnitude in the direction opposite to the direction of v = i j k. The vector we are looking for is v v. We have Therefore, v = 4 + 4 + 4 =.

More information

### 9-1 Similar Right Triangles (Day 1) 1. Review:

9-1 Similar Right Triangles (Day 1) 1. Review: Given: ACB is right and AB CD Prove: ΔADC ~ ΔACB ~ ΔCDB. Statement Reason 2. In the diagram in #1, suppose AD = 27 and BD = 3. Find CD. (You may find it helps

More information

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

More information

### Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

More information

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

More information

### How is a vector rotated?

How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 61-68 (1999) Introduction In an earlier series

More information

### MATH 150 TOPIC 9 RIGHT TRIANGLE TRIGONOMETRY. 9a. Right Triangle Definitions of the Trigonometric Functions

Math 50 T9a-Right Triangle Trigonometry Review Page MTH 50 TOPIC 9 RIGHT TRINGLE TRIGONOMETRY 9a. Right Triangle Definitions of the Trigonometric Functions 9a. Practice Problems 9b. 5 5 90 and 0 60 90

More information

### Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics.

MATH19730 Part 1 Trigonometry Section2 Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics. An angle can be measured in degrees

More information

### 7-6 The Law of Sines

7-6 The Law of Sines So far, we have learned how to use geometric mean, Pythagorean Theorem, properties of 30-60-90 and 45-45-90 triangles, and Soh, Cah, Toa to solve triangles. The Law of Sines is used

More information

### List of trigonometric identities

List of trigonometric identities From Wikipedia, the free encyclopedia In mathematics, trigonometric identities are equalities involving trigonometric functions that are true for all values of the occurring

More information

### Vectors-Algebra and Geometry

Chapter Two Vectors-Algebra and Geometry 21 Vectors A directed line segment in space is a line segment together with a direction Thus the directed line segment from the point P to the point Q is different

More information

### Trigonometric Identities and Equations

Chapter 4 Trigonometric Identities and Equations Trigonometric identities describe equalities between related trigonometric expressions while trigonometric equations ask us to determine the specific values

More information

### 1 Symmetries of regular polyhedra

1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

More information

### Construction and Properties of the Icosahedron

Course Project (Introduction to Reflection Groups) Construction and Properties of the Icosahedron Shreejit Bandyopadhyay April 19, 2013 Abstract The icosahedron is one of the most important platonic solids

More information

### 7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

More information

### Solution: 2. Sketch the graph of 2 given the vectors and shown below.

7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit

More information

### 4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.

14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right

More information

### 6.5 Trigonometric formulas

100 CHAPTER 6. TRIGONOMETRIC FUNCTIONS 6.5 Trigonometric formulas There are a few very important formulas in trigonometry, which you will need to know as a preparation for Calculus. These formulas are

More information

### M344 - ADVANCED ENGINEERING MATHEMATICS Lecture 9: Orthogonal Functions and Trigonometric Fourier Series

M344 - ADVANCED ENGINEERING MATHEMATICS ecture 9: Orthogonal Functions and Trigonometric Fourier Series Before learning to solve partial differential equations, it is necessary to know how to approximate

More information

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information