Three%20Dimensional%20Space

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Three Dimensional Space

• Dr. Farhana Shaheen
• Assistant Professor
• YUC-SA

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3-D Picture
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3-D
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3-D
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3-D Illusions
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Three-dimensional space

• Three-dimensional space is a geometric model of
  the physical universe in which we live. The three
  dimensions are commonly called length, width, and
  depth (or height), although any three mutually
  perpendicular directions can serve as the three
  dimensions.

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Space figures

• Space figures are figures whose points do not all
  lie in the same plane. We will discuss the
  polyhedron, the cylinder, the cone, and the
  sphere.
• Polyhedrons are space figures with flat surfaces,
  called faces, which are made of polygons. Prisms
  and pyramids are examples of polyhedrons.
• Cylinders, cones, and spheres are not
  polyhedrons, because they have curved, not flat,
  surfaces.
• A cylinder has two parallel, congruent bases that
  are circles.
• A cone has one circular base and a vertex that is
  
• not on the base.
• A sphere is a space figure having all its points
  an equal distance from the center point.

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Polygons Triangles,Squares, Pentagons

• Three-dimensional geometry, or space geometry, is
  used to describe the buildings we live and work
  in, the tools we work with, and the objects we
  create. First, we'll look at some types of
  polyhedrons. A polyhedron is a three-dimensional
  figure that has polygons as its faces. Its name
  comes from the Greek "poly" meaning "many," and
  "hedra," meaning "faces." The ancient Greeks in
  the 4th century B.C. were brilliant geometers.
  They made important discoveries and consequently
  they got to name the objects they discovered.
  That's why geometric figures usually have Greek
  names!

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PRISM

• We can relate some polyhedrons--and other space
  figures as well--to the two-dimensional figures
  that we're already familiar with. For example, if
  you move a vertical rectangle horizontally
  through space, you will create a rectangular or
  square prism.

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TRIANGULAR PRISM

• If you move a vertical triangle horizontally, you
  generate a triangular prism. When made out of
  glass, this type of prism splits sunlight into
  the colors of the rainbow.

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Prism splitting sunlight into rainbow colours
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Platonic Solids

• Platonic Solids are polyhedrons where all the
  faces are regular polygons, and all the corners
  have same number of faces joining them, and all
  the faces are exactly the same size and shape.

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Tetrahedron

• In geometry, a tetrahedron (plural tetrahedra)
  is a polyhedron composed of four triangular
  faces, three of which meet at each vertex. A
  regular tetrahedron is one in which the four
  triangles are regular, or "equilateral", and is
  one of the Platonic solids. The tetrahedron is
  the only convex polyhedron that has four faces. A
  tetrahedron is also known as a triangular
  pyramid.

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Tetrahedron
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Hexahedron
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Octahedron and Dodecahedron
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CONE

• A Cone is another familiar space figure with many
  applications in the real world.
• A cone can be generated by twirling a right
  triangle around one of its legs.
• If you like ice cream, you're no doubt familiar
  with at least one of them!

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Cone Shells and Cone Ice Cream
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CYLINDER

• Now let's look at some space figures that are not
  polyhedrons, but that are also related to
  familiar two-dimensional figures. What can we
  make from a circle? If you move the center of a
  circle on a straight line perpendicular to the
  circle, you will generate a cylinder. You know
  this shape--cylinders are used as pipes, columns,
  cans, musical instruments, and in many other
  applications.

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Cylindrical cans and flute
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SPHERE

• A sphere is created when you twirl a circle
  around one of its diameters. This is one of our
  most common and familiar shapes--in fact, the
  very planet we live on is an almost perfect
  sphere! All of the points of a sphere are at the
  same distance from its center.

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Spherical Shapes
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Rhombicosidodecahedron"?

• There are many other space figures--an endless
  number, in fact. Some have names and some don't.
  Have you ever heard of a "rhombicosidodecahedron"?
  Some claim it's one of the most attractive of
  the 3-D figures, having equilateral triangles,
  squares, and regular pentagons for its surfaces.
  Geometry is a world unto itself, and we're just
  touching the surface of that world.

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Rhombicosidodecahedron

• The 3-D figure, having
• equilateral triangles,
• squares, and regular
• pentagons
• for its surfaces.

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Rhombicosidodecahedrons
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Dimension

• In mathematics and physics, the dimension of a
  space or object is informally defined as the
  minimum number of coordinates needed to specify
  each point within it. Thus a line has a dimension
  of one because only one coordinate is needed to
  specify a point on it. A surface such as a plane
  or the surface of a cylinder or sphere has a
  dimension of two because two coordinates are
  needed to specify a point on it (for example, to
  locate a point on the surface of a sphere you
  need both its latitude and its longitude). The
  inside of a cube, a cylinder or a sphere is
  three-dimensional because three co-ordinates are
  needed to locate a point within these spaces.

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A diagram showing the first four spatial
dimensions
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• A drawing of the first four dimensions
• On the left is zero dimensions (a point) and on
  the right is four dimensions (a tesseract). There
  is an axis and labels on the right and which
  level of dimensions it is on the bottom. The
  arrows alongside the shapes indicate the
  direction of extrusion.

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• Below from left to right, is a square, a cube,
  and a tesseract.
• The square is bounded by 1-dimensional lines, the
  cube by 2-dimensional areas, and the tesseract by
  3-dimensional volumes.
• A projection of the cube is given since it is
  viewed on a two-dimensional screen. The same
  applies to the tesseract, which additionally can
  only be shown as a projection even in
  three-dimensional space.

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• Three dimensional Cartesian coordinate system
  with the x-axis pointing towards the observer

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Analytic geometry

• In mathematics, analytic geometry (also called
  Cartesian geometry) describes every point in
  three-dimensional space by means of three
  coordinates. Three coordinate axes are given,
  each perpendicular to the other two at the
  origin, the point at which they cross. They are
  usually labeled x, y, and z. Relative to these
  axes, the position of any point in
  three-dimensional space is given by an ordered
  triple of real numbers, each number giving the
  distance of that point from the origin measured
  along the given axis, which is equal to the
  distance of that point from the plane determined
  by the other two axes.

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Euclidean Space

• In mathematics, solid geometry was the
  traditional name for the geometry of
  three-dimensional Euclidean space for practical
  purposes the kind of space we live in. It was
  developed following the development of plane
  geometry. Stereometry deals with the measurements
  of volumes of various solid figures cylinder,
  circular cone, truncated cone, sphere, prisms,
  blades, wine casks.

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Three Dimensional Plane
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Cylindrical and Spherical Coordinates

• Other popular methods of describing the location
  of a point in three-dimensional space include
  cylindrical coordinates and spherical
  coordinates, though there are an infinite number
  of possible methods.

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• A cylindrical coordinate system is a
  three-dimensional coordinate system, where each
  point is specified by the two polar coordinates
  of its perpendicular projection onto some fixed
  plane, and by its (signed) distance from that
  plane.
• Cylindrical coordinates are useful in connection
  with objects and phenomena that have some
  rotational symmetry about the longitudinal axis,
  such as water flow in a straight pipe with round
  cross-section, heat distribution in a metal
  cylinder, etc.

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• A cylindrical coordinate system with origin O,
  polar axis A, and longitudinal axis L. The dot is
  the point with radial distance ?  4, angular
  coordinate f  130, and height z  4.

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Spherical Coordinates (r, ?, F)
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Sherical Coordinates (r, ?, F)
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• Spherical coordinates, also called spherical
  polar coordinates, are a system of curvilinear
  coordinates that are natural for describing
  positions on a sphere or spheroid. Define ? to be
  the azimuthal angle in the xy-plane from the
  x-axis ,
• F to be the polar angle (also known as the
  zenith angle , and ? to be distance (radius) from
  a point to the origin. This is the convention
  commonly used in mathematics.
• SPHERICAL COORDINATES

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Set of points where ? (rho) is constant
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Points where F (phi) is constant
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Points where ? (theta) is constant
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• What happens when they are all constant
• (one by one)
• (?, ?, F) (Rrho, Pphi, Ttheta)

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Viewing Three Dimensional Space

• Another mathematical way of viewing
  three-dimensional space is found in linear
  algebra, where the idea of independence is
  crucial. Space has three dimensions because the
  length of a box is independent of its width or
  breadth. In the technical language of linear
  algebra, space is three dimensional because every
  point in space can be described by a linear
  combination of three independent vectors. In this
  view, space-time is four dimensional because the
  location of a point in time is independent of its
  location in space.

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Viewing Three Dimensional Space

• Three-dimensional space has a number of
  properties that distinguish it from spaces of
  other dimension numbers. For example, at least 3
  dimensions are required to tie a knot in a piece
  of string. Many of the laws of physics, such as
  the various inverse square laws, depend on
  dimension three.
• The understanding of three-dimensional space in
  humans is thought to be learned during infancy
  using unconscious inference, and is closely
  related to hand-eye coordination. The visual
  ability to perceive the world in three dimensions
  is called depth perception.

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Skew Lines

• In solid geometry, skew lines are two lines that
  do not intersect but are not parallel.
  Equivalently, they are lines that are not both in
  the same plane. A simple example of a pair of
  skew lines is the pair of lines through opposite
  edges of a regular tetrahedron (or other
  non-degenerate tetrahedron). Lines that are
  coplanar either intersect or are parallel, so
  skew lines exist only in three or more
  dimensions.

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Example of skew lines
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Four Skew Lines
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Everyday example of Skew lines
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Skew lines in Air Shows
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Air displays
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Graphs in 2-D
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Surfaces in 3 - D
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3-D surface
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Example of function of two variable

• So far, we have dealt with functions of single
  variables only. However, many functions in
  mathematics involve 2 or more variables. In this
  section we see how to find derivatives of
  functions of more than 1 variable.
• Here is a function of 2 variables, x and y
• F(x,y) y 6 sin x 5y2
• To plot such a function we need to use a
  3-dimensional co-ordinate system.

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F(x,y) y 6 sin x 5y2
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To find Partial Derivatives

• "Partial derivative with respect to x" means
  "regard all other letters as constants, and just
  differentiate the x parts".
• In our example (and likewise for every 2-variable
  function), this means that (in effect) we should
  turn around our graph and look at it from the far
  end of the y-axis. So we are looking at the x-z
  plane only.

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Graph of F(x,y) y 6 sin x 5y2taking y
constant
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We see a sine curve at the bottom and this
comes from the 6 sin x part of our
functionF(x,y) y 6 sin x 5y2. The y parts
are regarded as constants.
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• (The sine curve at the top of the graph is just
  where the software is cutting off the surface -
  it could have been made it straight.)
• Now for the partial derivative of
• F(x,y) y 6 sin x 5y2
• with respect to x

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Partial Differentiation with respect to y

• "Partial derivative with respect to y" means
  "regard all other letters as constants, just
  differentiate the y parts".
• As we did above, we turn around our graph and
  look at it from the far end of the x-axis. So we
  see (and consider things from) the y-z plane
  only.

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Graph of F(x,y) y 6 sin x 5y2 taking x
constant
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Parabola

• We see a parabola. This comes from the y2 and y
  terms in
• F(x,y) y 6 sin x 5y2.
• The "6 sin x" part is now regarded as a constant.

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• Now for the partial derivative of
• F(x,y) y 6 sin x 5y2
• with respect to y.

The derivative of the y-parts with respect to y
is 1 10y. The derivative of the 6 sin x part is
zero since it is regarded as a constant when we
are differentiating with respect to y.
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