Elliptic Geometry_LISBE-ELCA-MEIRA-final.pptx

Elliptic Geometry

1. ELLIPTIC GEOMETRY Reporter: Elca Meira C. Lisbe PRESENTATION TITLE

2. Learning objectives Define Elliptic Geometry Differentiate Elliptic Geometry to Euclidean Geometry Differentiate between single and double elliptic geometry Define terms related to elliptic geometry Explain the basic facts of Elliptic Geometry Learn the application of the concepts of elliptic geometry to real-life situations

3. EUCLID’S PARALLEL POSTULATE

4. Problem Non- Euclidean Geometry geometry that violates the parallel principle PRESENTATION TITLE Euclidean Geometry geometry of flat surfaces

5. Solution Our product makes consumer lives easier, and no other product on the market offers the same features​ Gen Z (18-25 years old) Reduce expenses for replacement products  Simple design that gives customers the targeted information they need​ PRESENTATION TITLE Elliptical Geometry geometry of spherical surfaces Also called Riemannian Geometry Named after German Mathematician, Bernhard Riemann Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°.

6. Two Variations of Elliptic Geometry Single Elliptic Geometry (Elliptic Geometry) Double Elliptic Geometry each pair of lines intersect in exactly one point each pair of lines intersect in exactly two points hemispherical model is used Spherical model is used Both have no parallel lines In both types, straight lines are finite in length.

7. We based our research on market trends and social media ​ We believe people need more products specifically dedicated to this niche market ​ Minimalist and easy to use  ​ PRESENTATION TITLE Double Elliptic Geometry In the spherical model, the surface of a sphere represents all the points in the geometry. A great circle is the circular intersection of a sphere and a plane passing through the sphere’s center point. If the radius of the sphere is r, then the length of each great circle is   Two great circles meet in two points known as antipodal points . (points at the ends of a diameter of the sphere)

8. We based our research on market trends and social media ​ Minimalist and easy to use  ​ PRESENTATION TITLE Facts about Points and Straight Lines (great circles) Each pair of straight lines meet in two points known as antipodal points. Through each pair of antipodal points passes an infinite number of straight lines. Through each pair on n on-antipodal points passes a unique straight line. Through each point there pass infinitely many straight lines, the totality of which covers the entire sphere.

9. Product Benefits Cool and stylish product​ Areas for community connections Online store and market swap​ PRESENTATION TITLE 10 Minimalist and easy to use  ​ PRESENTATION TITLE The shortest path on a sphere joining one point to another is the geodesic arc . For non-antipodal points, there is a second, longer path between the points. For antipodal points, there are two shortest paths between the points. A geodesic arc is always an arc of a unique great circle. A geodesic arc is similar to the line segment in Euclidean geometry.

10. Product Benefits Cool and stylish product​ Areas for community connections Online store and market swap​ PRESENTATION TITLE 11 Minimalist and easy to use  ​ PRESENTATION TITLE The distance between two non-antipodal points on a sphere is the length of the geodesic arc joining them. If two points are antipodal, then the distance between them is half the circumference of a great circle  

11. More Facts about Double Elliptic Geometry If A, B, C are any three points on a sphere, then . This relationship between three points is known as the triangle inequality. The sum of the angles of a spherical triangle is greater than and less than . Two spherical triangles are said to be congruent if their corresponding sides and angles are equal. The triangle congruence principles like SSS, SAS, ASA and AAA hold true in a spherical geometry. While the principle AAS triangle congruence does not necessary hold true.  

12. Growth strategy How we’ll scale in the future Roll out product to high profile or top-level participants to help establish the product​ Release the product to the general public and monitor press release and social media accounts​ Gather feedback and adjust product design as necessary​ PRESENTATION TITLE Area of a spherical triangle: Where: r = radius of the sphere = measures of the angles ( in radians )  

13. Growth strategy How we’ll scale in the future Roll out product to high profile or top-level participants to help establish the product​ Release the product to the general public and monitor press release and social media accounts ​ Gather feedback and adjust product design as necessary​ PRESENTATION TITLE Napier’s Rules for Right S pherical T riangles For a right triangle on a sphere of radius r (as shown in the figure): Similar formulas hold for the angle It is possible for a triangle to have 0, 1, 2, or 3 right angles in spherical geometry.  

14. Growth strategy How we’ll scale in the future Roll out product to high profile or top-level participants to help establish the product​ Release the product to the general public and monitor press release and social media accounts ​ Gather feedback and adjust product design as necessary​ PRESENTATION TITLE Napier’s Rules for Right S pherical T riangles For a right triangle on a sphere of radius r (as shown in the figure): Similar formulas hold for the angle It is possible for a triangle to have 0, 1, 2, or 3 right angles in spherical geometry.  

15. Product Overview PRESENTATION TITLE Spherical Law of Sines Spherical Law of Cosines Note: There are many more formula for spherical geometry besides these

16. Business Model We based our research on market trends and social media ​ We believe people need more products specifically dedicated to this niche market ​ Minimalist and easy to use  ​ PRESENTATION TITLE More Facts about Double Elliptic Geometry All the great circles perpendicular to a given great circle A , meet in two antipodal points known as poles of A . In the figure, the great circles B and C , perpendicular to great circle A meet at poles N and S . You can think of A as being the equator on a globe, the great circles perpendicular to A as longitudinal (aka meridian lines), and N and S as being the North and South poles.

17. Mirjam Nilsson​​ 206-555-0146​ [email protected]​ www.contoso.com​ PRESENTATION TITLE 18 What is the use of Elliptic Geometry? As the Earth is not flat, the shortest distance between two places is usually not a straight line that is drawn on a flat map. Elliptic geometry can be used to calculate the shortest distance between places that are far away from each other.

18. PRESENTATION TITLE airlines and pilots can use it to calculate the best route between two cities , and ship captains can use it to calculate the best route between two harbors.

19. Elliptic geometry is also used in space exploration and cosmology

20. Our team PRESENTATION TITLE Sources https://study.com/academy/lesson/video/elliptic-geometry-definition-postulates.html Coxeter , H.S.M, Introduction to Geometry, John Wiley & Sons, 1969 Gans , D. (1973). Introduction to non-Euclidean geometry (Illustrated ed., Vol. 9). Elsevier Science https://www.youtube.com/watch?v=ofoYU6y3fR4&list=PPSV Note: Photo sources are placed in the Notes section in each slide

21. Our team PRESENTATION TITLE Q&A: Let’s Dive Deeper 1.) Show by an example that an exterior angle of a spherical triangle may equal an opposite interior angle of the triangle . 2.) Given that a point of a geodesic arc divides it into two arcs, prove that they are also geodesic arcs.

22. PRESENTATION TITLE Thank you for listening!