PlANE, SOLID AND COORDINATE GEOMETRY

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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

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• Plane Geometry

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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.

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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

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Axioms of Euclids 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.

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Common Notions (Axioms)

1. Things that equal the same thing also equal one
   another.
2. If equals are added to equals, then the wholes
   are equal.
3. If equals are subtracted from equals, then the
   remainders are equal.
4. Things that coincide with one another equal one
   another.
5. The whole is greater than the part.

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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

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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.

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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.

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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.
•  

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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..

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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.

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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.

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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
•  

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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

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Question

• State whether the figures below are polygons or
  not ?
• a.
  b.

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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.

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Regular Polygon

• 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.

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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.

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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.

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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
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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.

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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,

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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."
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• Coordinate
• Geometry

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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.

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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).

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Converting Polar and Cartesian coordinates

• To convert from Cartesian Coordinates (x,y) to
  Polar Coordinates (r,?)
• To convert from Polar coordinates (r, ?) to
  Cartesian coordinates

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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

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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

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Further Terminology
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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.

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Ellipse cont

• Ellipses with Horizontal Major Axis
• Ellipses with Vertical Major Axis

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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.

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Hyperbola cont..

• Horizontal transverse axis
• Vertical transverse axis

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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 P2a, 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.
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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.

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Two-dimensional spirals
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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.

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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.

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• Solid Geometry

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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.

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• 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

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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).

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Hyperboloid

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

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Plot 3d Figures in Matlab
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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.

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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.

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The 13 Archimedean Solids
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• Further Topics in Geometry

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Euclids 5th Postulate

• That if a straight line falling on two straight
  lines makes the interior angles less that two
  right angles, the two straight lines, if produced
  indefinitely , meet on that side on which the
  angles are less than tow right angles.
• In other words Through an exterior point of a
  straight line ( a line not on the straight line)
  one can construct one and only parallel to the
  given straight line.
• The 5th Postulate is logically consistent in
  itself and forms an axiomatic system with the
  other 4 postulates.

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• But while forming an axiomatic system, the 5th
  postulate was thought to be dependant on the
  first 4.
• Therefore mathematicians through out the past,
  redefined the 5th postulate with new theories
  and gave way to non-Euclidian geometry. E.g.
  Hyperbolic geometry, Elliptic geometry.

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Non-Euclidian Geometry

• The axioms of Geometry were formerly regarded as
  laws of thought which an intelligent mind could
  neither deny nor investigate.
• However, that it is possible to take a set of
  axioms, wholly or in part contradicting those of
  Euclid, and build up a Geometry as consistent as
  his.
• Examples of non-Euclidean geometries include the
  hyperbolic and elliptic geometry, which are
  contrasted with a Euclidean geometry.
• The essential difference between Euclidean and
  non-Euclidean geometry is the nature of parallel
  lines.

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• Another way to describe the differences between
  these geometries is to consider two straight
  lines indefinitely extended in a two-dimensional
  plane that are both perpendicular to a third
  line
• In Euclidean geometry the lines remain at a
  constant distance from each other, and are known
  as parallels.
• In hyperbolic geometry they "curve away" from
  each other, increasing in distance as one moves
  further from the points of intersection with the
  common perpendicular these lines are often
  called ultra parallels.
• In elliptic geometry the lines "curve toward"
  each other and eventually intersect.

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Triangles in different spaces
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Hyperbolic Geometry

• Hyperbolic geometry (also called Lobachevskian
  geometry) was created in the first half of the
  nineteenth century in the midst of attempts to
  understand Euclid's axiomatic basis for geometry.
• It is one type of non-Euclidean geometry that
  discards Euclid's 5th postulate.
• It is replaced by the postulate which states
  that Given a line and a point not on it, there
  is more than one line (infinitely many lines)
  going through the given point that is parallel to
  the given line.

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• The parallel postulate in Euclidean geometry is
  equivalent to the statement that, in two
  dimensional space, for any given line l and point
  P not on l, there is exactly one line through P
  that does not intersect l i.e., that is parallel
  to l. In hyperbolic geometry there are at least
  two distinct lines through P which do not
  intersect l, so the parallel postulate is false.

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• An example of such a case in hyperbolic geometry
  , is the hyperbola. Where the hyperbola, though
  it approaches the asymptote it never meets
  it.(This violates Euclids parallel postulate)
• Applications of hyperbolic geometry includes
  topics such as Toplogy, Group Theory and Complex
  variables conformal mapping.

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Problems unsolved in geometry.

• The Hadwiger problem.
• The Polygonal illumination problem.
• The Chromatic Number of the plane.
• Kissing Numbers.??
• Perfect cuboids.
• The Kabon Triangle Problem??
• There are many more.. Google and explore!!!

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Kissing numbers

• In d dimensions, the kissing number K(d) is the
  maximum number of disjoint unit spheres that can
  touch a given sphere.
• What could be K(2) and K(3)??

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Kabon triangle problem

• The problem is to find how many disjoint
  triangles can be created with n lines in the
  plane (K(n))
• What could be the sequence of K(n) ??

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At the end of this lecture

• We hope you would have been enlightened about
  the broader perspective of geometry. Namely
  plane, solid and coordinate geometry.
• You would have realized the need for Mathematical
  thinking and reasoning!!
• We also hope you would take the formulae of
  different curves in your minds and apply them
  when you come across mathematical problems.
• Please go through any new words you came
  across..!!

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• The End