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History and Philosophy of Mathematics MA0010

History and Philosophy of Mathematics MA0010. PlANE, SOLID AND COORDINATE GEOMETRY Conducted by Department of Mathematics University of MORATUWA Ms Shanika FeRDiNANDIS Mr. Kevin Rajamohan. Plane Geometry. Euclid ( Father of Geometry). Euclidean Geometry

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History and Philosophy of Mathematics MA0010

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  1. History and Philosophy of Mathematics MA0010 PlANE, SOLID AND COORDINATE GEOMETRY Conducted by Department of Mathematics University of MORATUWA Ms Shanika FeRDiNANDIS Mr. Kevin Rajamohan

  2. Plane Geometry Department of Mathematics,UOM

  3. Euclid ( Father of Geometry) Euclidean Geometry • Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. • The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. Department of Mathematics,UOM

  4. Some basic results in Euclidean Geometry • The sum of angles A, B, and C is equal to 180 degrees. • The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). • Thales' theorem: if AC is a diameter then the angle at B is a right angle Department of Mathematics,UOM

  5. Axioms of Euclid’s Geometry • Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated: • [It is possible] to draw a straight line from any point to any point. • [It is possible] To produce [extend] a finite straight line continuously in a straight line. • [It is possible] To describe a circle with any center and distance [radius]. • That all right angles are equal to one another. • The parallel postulate: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Department of Mathematics,UOM

  6. Common Notions (Axioms) • Things that equal the same thing also equal one another. • If equals are added to equals, then the wholes are equal. • If equals are subtracted from equals, then the remainders are equal. • Things that coincide with one another equal one another. • The whole is greater than the part. Department of Mathematics,UOM

  7. Nine point circle • The nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points, six lying on the triangle itself (unless the triangle is obtuse). They include: • The midpoint of each side of the triangle. • The foot of each altitude. • The midpoint of the segment of each altitude from its vertex to the orthocenter (where the three altitudes meet). Department of Mathematics,UOM

  8. Centroid • The centroid (G) of a triangle is the common intersection of the three medians of a triangle. A median of a triangle is the segment from a vertex to the midpoint of the opposite side. Department of Mathematics,UOM

  9. Orthocenter The orthocenter (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side. Department of Mathematics,UOM

  10. Circumcenter The circumcenter (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by two points, (C) is on the perpendicular bisector of each side of the triangle. Note (C) may be outside the triangle. Department of Mathematics,UOM

  11. Euler Line In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral; it passes through several important points determined from the triangle. In the image, the Euler line is shown in red. It passes through the orthocenter (blue), the circumcenter (green), the centroid (orange), and the center of the nine-point circle (red) of the triangle.. Department of Mathematics,UOM

  12. Pythagorean Theorem: Different Proofs • This is a theorem that may have more known proofs than any other; the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs. • Proof using similar triangles • Let ABC represent a right angle triangle. • Draw an altitude from point C and call H its intersection with the side AB. • The new triangle ACH is similar to ABC. (by definition of the altitude, they both have a right angle) • Similarly, triangle CBH is similar to ABC. Department of Mathematics,UOM

  13. Proof using similar triangles cont… • The similarities lead to the two ratios: • These can be written as • Summing these two equalities, we obtain • In other words, The Pythagorean theorem: • Exercise: Prove the Pythagorean theorem in one other way. Department of Mathematics,UOM

  14. The Pythagorean Theorem in 3D • The Pythagorean Theorem, which allows you to find the hypotenuse of a right triangle, can also be used in three dimensions to find the diagonal length of a rectangular prism. This is the distance d  from one corner of the box to the furthest opposite corner, as shown in the diagram at the right. • The distance can be calculated using: Department of Mathematics,UOM

  15. Polygons • In geometry a polygon is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners. • The following are examples of polygons: Department of Mathematics,UOM

  16. Question: • State whether the figure’s below are polygons or not ? a. b. Department of Mathematics,UOM

  17. Vertex • The vertex of an angle is the point where the two rays that form the angle intersect. • The vertices of a polygon are the points where its sides intersect. Department of Mathematics,UOM

  18. RegularPolygon • A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same. The sum of the angles of a polygon with n sides, where n is 3 or more, is 180° × (n - 2) degrees. Department of Mathematics,UOM

  19. Triangle- Three sided polygon • Equilateral Triangle or Equiangular Triangle A triangle having all three sides of equal length. The angles of an equilateral triangle all measure 60 degrees. • Isosceles Triangle A triangle having two sides of equal length. • Right Triangle A triangle having a right angle. One of the angles of the triangle measures 90 degrees. The side opposite the right angle is called the hypotenuse. Department of Mathematics,UOM

  20. Four sided Polygons • Parallelogram A four-sided polygon with two pairs of parallel sides. • Rhombus A four-sided polygon having all four sides of equal length. • Trapezoid A four-sided polygon having exactly one pair of parallel sides. The two sides that are parallel are called the bases of the trapezoid. Department of Mathematics,UOM

  21. Tessellation A Tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Only three regular polygons tessellate in the Euclidean Plane: Triangles, Squares or Hexagons. A tessellation of triangles A tessellation of squares A tessellation of hexagons Department of Mathematics,UOM

  22. Compass and straightedge • Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass. • Every point constructible using straightedge and compass may be constructed using compass alone. A number of ancient problems in plane geometry impose this restriction. Department of Mathematics,UOM

  23. Trisecting an angle • Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836) French mathematician. Angles may not in general be trisected • The geometric problem of angle trisection can be related to algebra – specifically, the roots of a cubic polynomial – since by the triple-angle formula, Department of Mathematics,UOM

  24. Gauss Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy and optics. Sometimes known as the “the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. He referred to mathematics as "the queen of sciences." Department of Mathematics,UOM

  25. Coordinate Geometry Department of Mathematics,UOM

  26. Coordinate Geometry Cartesian Coordinates • In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represented by a pair of numbers (x,y). • x is the signed distance from the y-axis to the point P, and • y is the signed distance from the x-axis to the point P. • In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represented by a triple of numbers (x,y,z). • x is the signed distance from the yz-plane to the point P, • y is the signed distance from the xz-plane to the point P, and • z is the signed distance from the xy-plane to the point P. Department of Mathematics,UOM

  27. Coordinate Geometry Polar Coordinates The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles. They are the most common systems of curvilinear coordinates. The term polar coordinates often refers to circular coordinates (two-dimensional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both three-dimensional). Department of Mathematics,UOM

  28. Converting Polar and Cartesian coordinates To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ): To convert from Polar coordinates (r, θ) to Cartesian coordinates Department of Mathematics,UOM

  29. Circle A circle is the set of points in a plane that are equidistant from a given point . The distance from the center r is called the radius, and the point o is called the center. Twice the radius is known as the diameter . In Cartesian coordinates, the equation of a circle of radius r centered on (h,k) is Department of Mathematics,UOM

  30. Area of a Circle • This derivation was first recorded by Archimedes in Measurement of a Circle (ca. 225 BC). • If the circle is instead cut into wedges, as the number of wedges increases to infinity, a rectangle results, so Department of Mathematics,UOM

  31. Further Terminology Department of Mathematics,UOM

  32. Ellipse • The ellipse is defined as the locus ( A the set of all points satisfying some condition) of a point (x,y) which moves so that the sum of its distances from two fixed points (called foci, or focuses ) is constant. Department of Mathematics,UOM

  33. Ellipse cont… • Ellipses with Horizontal Major Axis • Ellipses with Vertical Major Axis Department of Mathematics,UOM

  34. Hyperbola The word "hyperbola" derives from the Greek meaning "over-thrown" or "excessive", from which the English term hyperbole derives. In mathematics a hyperbola is a smooth planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bows aimed at each other. Department of Mathematics,UOM

  35. Hyperbola cont.. • Horizontal transverse axis • Vertical transverse axis Department of Mathematics,UOM

  36. Parabola A parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by P=2a, where a is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a parabolid. Department of Mathematics,UOM

  37. Spiral • A spiral is typically a planar curve (that is, flat), like the groove on a record or the arms of a spiral galaxy. • A spiral emanates from a central point, getting progressively farther away as it revolves around the point. Department of Mathematics,UOM

  38. Two-dimensional spirals Department of Mathematics,UOM

  39. Cycloid • A cycloid is the locus of a point on the rim of a circle of radius a rolling along a straight line. The cycloid was first studied by Cusa when he was attempting to find the area of a circle by integration. It was studied and named by Galileo in 1599. Department of Mathematics,UOM

  40. Hypocycloid • The path traced out by a point on the edge of a circle of radius b rolling on the outside of a circle of radius a. Department of Mathematics,UOM

  41. Solid Geometry Department of Mathematics,UOM

  42. Sphere • Spherical surface has been defined as the locus of points in three-dimensional space, at a given distance from a given point. • The given point is called the center. The given distance is called a radius. • Sphere is a solid bounded by a spherical surface. Department of Mathematics,UOM

  43. In analytic geometry, a sphere with center (a, b, c) and radius r is the locus of all points (x, y, z) such that • Refer on the Properties of the sphere. • The points on the sphere with radius r can be parameterized by Department of Mathematics,UOM

  44. Ellipsoid • An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is • Where a and b are the equatorial radii (along the x and y axes) and c is the polar radius (along the z-axis). Department of Mathematics,UOM

  45. Hyperboloid • A hyperboloid is a type of surface in three dimensions, described by the equation • Refer the importance of Hyperboloid structures in Construction engineering. Department of Mathematics,UOM

  46. Plot 3d Figures in Matlab Department of Mathematics,UOM

  47. Platonic solids • Tetrahedron, Cube, Octahedron, Dodecahedron & Icosahedron – These 5 solids are called Perfect solids or Platonic solids (in which a constant number of identical regular faces meet at each vertex) • They are known as Perfect, because of their unique construction-They are the only forms we know of, that have multiple sides which all have the same shapes & size. Department of Mathematics,UOM

  48. Archimedean Polyhedra • They are formed from Platonic Solids by cutting off the corners ( Truncated Polyhedra). • It is a solid made out of, more than one polygon. • All the vertices are identical. Department of Mathematics,UOM

  49. The 13 Archimedean Solids Department of Mathematics,UOM

  50. Further Topics in Geometry Department of Mathematics,UOM

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