Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.

Slides:

Advertisements

Similar presentations

Polygon from a known side

Advertisements

Circle Theorems-“No Brainers”

Tangents, Arcs, and Chords

Geometric Construction

Chapter 11 GRUDGE REVIEW.

Introduction Construction methods can also be used to construct figures in a circle. One figure that can be inscribed in a circle is a hexagon. Hexagons.

Constructing Regular Hexagons Inscribed in Circles Adapted from Walch Education.

Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve. Consider the region.

10.3 Inscribed Angles Goal 1: Use inscribed angles to solve problems Goal 2: Use properties of inscribed polygons CAS 4, 7, 16, 21.

Figures and Shapes Northern Computer Lab.

Section 5.2 – Central Angles and Arcs Objective To find the length of an arc, given the central angle Glossary Terms Arc – a part of a circle Central angle.

By Abdul Chebly.  obtuse angle- an angle that’s less then 180 and higher then 90 degrees.  Right angle- an angle that is 90 degrees.

8.4 Areas of Regular Polygons “If I had to live my life again, I’d make the same mistakes, only sooner.” Tallulah Bankhead.

9-2 Area of Regular Polygons The center of a regular polygon is equidistant from its vertices. Radius- the distance from the center to a vertex. Apothem-

Areas and Volumes. Area of a circle We need a substitution.

Section 8.4 Nack/Jones1 Section 8.4 Polyhedrons & Spheres.

Geometry Review AREA 1. Find the measure of each interior angle of the regular polygon shown below. 2.

Note 2: Perimeter The perimeter is the distance around the outside of a shape. Start at one corner and work around the shape calculating any missing sides.

Inscribed Angles. Inscribed Angles and Central Angles A Central angle has a vertex that lies in the center of a circle. A n inscribed angle has a vertex.

Lesson Handout #1-7 (ODD), (ODD) ** For each question:  BOX the exact value  CIRCLE the approximate value (.01)

Regular Polygons (all sides same length and all angles the same)  r² sounds like area to me, when I need the circumference I’ll just use  d CYLINDER.

Section 11-2 Areas of Regular Polygons. Area of an Equilateral Triangle The area of an equilateral triangle is one fourth the square of the length of.

Polygon in Annulus. You have two figures that each show a regular polygon (with a given side length) just touching two circles. What is the area of the.

Find the area of the triangle below. 3/24 with review 7.4 and 7.5 on 3/ Areas of Regular Polygons.

Multiple Integration Copyright © Cengage Learning. All rights reserved.

 Hidden Lines in Tessellations ◦ “Mind’s Eye” – the angle defined by our mind’s eye to help us find the pattern. ◦ Angles are all the same. ◦ These angles.

Polygons – Angles In Regular Polygons Regular Polygons have equal sides and equal angles, So if we can find one side, we would know the measure of all.

Geometry Review Lines, Angles, Polygons. What am I?

11.5 Sectors and Arc Lengths

Section 11-4 Areas of Regular Polygons. Given any regular polygon, you can circumscribe a circle about it.

Polygons and Triangles Chapter 10 Lesson 4 and 5

 Copy thm 12.9, corollaries, and thm from pgs 679 and 680.

9.5/9.6 Inscribed Angles and Quadrilaterals HW: 9.5 (#5 – 13) 9.6 (#9 – 13 )

Circles Modified by Lisa Palen. Definitions Circle The CENTER of the circle is the point that is the same distance to every point on the circle. The distance.

Clear your desk for the Quiz. Arc Length & Area Arc Length The length of a continuous curve r(θ) on the interval [  ] is equal to.

Objective: Measures of Inscribed Angles & Inscribed Polygons. (3.12.3) Section 10.4.

Warm-up Get out your homework. Agenda Review Homework Section 9-1, 9-2 Basic Circle Concepts Section 10-6 Geometric Probability Homework.

Francis González Shapes: 2 Dimensions Figures Rectangle, Square, Triangle, Circle, Arch 3 Dimensions Figures Cube, Pyramid, Sphere, Cylinder.

Objective: Finding interior and exterior angles of polygons. Midsegments. Warm up 1.Find the measure of angle J.

Chapter 9 Circles (page 328) How can relationships in a circle allow you to solve problems involving segments, angles, and arcs?

10-7 Area of Circles and Sectors. REVIEW Circumference: The total distance (in length) around a circle. Arc measure: The measure of the central angle.

9.2 Surface Area of Pyramids

Arcs, Sectors & Segments

7-7 Areas of Circles and Sectors

11.6 Arc Lengths and Areas of Sectors

A geometric shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric.

Introduction Triangles are not the only figures that can be inscribed in a circle. It is also possible to inscribe other figures, such as squares. The.

Rotational motion (rotary motion/circular motion/radial motion) An object that rotates about an axis of rotation through an angle q, over a distance s.

7-5 Areas of Regular Polygons

All the polygons in this game are regular polygons.

16.2 Arc Length and Radian Measure

11.3 Sector Area and Arc Length (Part 1)

5.2 Congruent Polygons.

Polygons – Angles In Regular Polygons

12-2 Arcs and Chords.

Arc Length and Sector Area

دانشگاه شهیدرجایی تهران

LESSON 7.5 AREAS OF REGULAR POLYGONS OBJECTIVE:

تعهدات مشتری در کنوانسیون بیع بین المللی

Lesson 2.6 Subsets of Space pp

7-5 Areas of Regular Polygons

DO NOW-Opportunity to get 5 points on test

Central Angles and Arc Length

11 Chapter Introductory Geometry

Parallelograms, Triangles, Rhombuses Rectangles & Trapezoids Regular

Inscribed Angles.

Lesson 8-2 Formulas Circumference Arc Length Area Sector.

8. Constructions and Loci

Module 15: Lesson 1 Interior & Exterior Angles

Presentation transcript:

Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) Distance sets on circles

Distance sets on spheres It is well-known that Def. A regular polygon attains this bound as t=1. A k-distance set on a circle lies on a regular polygon if k is small enough. Prob. Any DS with is.

Distance sets on regular polygons Regular Polygon The number of distances =The number of angles =The number of length of arcs ≒ (The number of differences as elements of )/2

Main Thm (M-Shinohara) Thm. Exampl e

The bound is sharp. Thm. This bound is sharp!

How to get main Thm 2. Distance sets on 1. Partition of the unit circle The number of distances =The number of length of arcs ∃ a line through the origin partitioning X into two parts of equal size.

How to get main Thm 2. Distance sets on 1. Partition of the unit circle 3. Fusion of two distance sets on 4. An application of Kneser’s addition Thm (When does it lie on regular polygons?) The number of distances =The number of length of arcs ∃ a line through the origin partitioning X into two parts of equal size. Prop.

4. An application of Kneser’s Thm Thm (Kneser, 1953). Cor.

4. An application of Kneser’s Thm Cor. Assume that. ・

Main Thm (M-Shinohara) Thm. Resul t. Thanks for your attention!