Post on 17-May-2015
description
Three Dimensional Space
Dr. Farhana ShaheenAssistant Professor
YUC-SA
3-D
3-D
3-D Illusions
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.
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.
A diagram showing the first four spatial dimensions
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.
Metatron's Cube
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.
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 (x, y, z) of real numbers x, y, z,
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.
Analytic geometry
Three dimensional Cartesian coordinate system with the x-axis pointing towards the observer.
Three Dimensional Plane
Shapes in 2-D and 3-D
Polyhedron CylinderCone
Sphere
Space figures are figures whose points do not all lie in the same plane.
For examples:
Space figures
Polyhedrons are space figures with flat surfaces, called faces, which are made of polygons.
Prisms and pyramids are examples of polyhedrons.
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.
Note that cylinders, cones, and spheres are not
polyhedrons, because they have curved, not flat, surfaces.
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!
Examples of polyhedrons
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.
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.
Prism splitting sunlight into rainbow colours
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.
Crystal Platonic Solids
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.
Icosahedron models
Dodecahedron
MerKaBa
Merkaba
The Merkaba is an extremely powerful symbol. It is a combination of two star tetrahedrons - one pointing up to the heavens, channelling energy down to the earth plain, and one pointing downwards, drawing up energy from the earth beneath. The top, or upward pointing tetrahedron is male and rotates clockwise, with the bottom or downwards pointing one being female, which rotates counter-clockwise
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!
Cone Shells and Cone Ice Cream
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.
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.
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.
Rhombicosidodecahedron
The 3-D figure, having equilateral triangles,squares, and regularpentagonsfor its surfaces.
Rhombicosidodecahedrons
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.
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.
Cylindrical coordinate system(ρ, φ, z)
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 ρφ-plane
and by its (signed) distance z from that plane.
Application of cylindrical Coordinates
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.
Example
A cylindrical coordinate system with origin O, polar axis A, and longitudinal axis L. The dot is the point with radial distance ρ = 4, angular coordinate φ = 130°, and height z = 4.
Spherical Coordinates (r, θ, Φ)
Spherical Coordinates (r, θ, Φ)
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 θ to be the azimuthal angle in the xy-plane from the x-axis ,
Φ to be the polar angle (also known as the zenith angle , and ρ to be distance (radius) from a point to the origin. This is the convention commonly used in mathematics.
SPHERICAL COORDINATES
Set of points where ρ (rho) is constant
Points where Φ (phi) is constant
Points where θ (theta) is constant
What happens when ρ, θ, Φ are all constant (one by one)
(ρ, θ, Φ) = (Rrho, Pphi, Ttheta)
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.
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.
Skew Lines
In solid geometry, skew lines are two lines that neither intersect nor are they 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.
Four Skew Lines
Skew lines in Air Shows
Air displays
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Graphs in 2-D
2D-Graphs
Surfaces in 3 - D
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.
F(x,y) = y + 6 sin x + 5y2
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.
Graph of F(x,y) = y + 6 sin x + 5y2
taking y constant
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.
(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 ofF(x,y) = y + 6 sin x + 5y2
with respect to x:
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.
Graph of F(x,y) = y + 6 sin x + 5y2 taking x constant
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.
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.