Three Dimensional Three Dimensional SpaceSpace

Dr. Farhana ShaheenAssistant Professor

YUC-SA

3-D Picture3-D Picture

3-D3-D

3-D3-D

3-D Illusions

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.

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.

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!

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 Prism splitting sunlight into rainbow colours rainbow colours

Platonic SolidsPlatonic 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.

TetrahedronTetrahedron

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.

TetrahedronTetrahedron

HexahedronHexahedron

Octahedron and Octahedron and DodecahedronDodecahedron

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.

Cylindrical cans and fluteCylindrical cans and flute

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.

Spherical ShapesSpherical Shapes

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

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 diagram showing the first four spatial dimensions

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.

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.

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

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.

Analytic geometry

Euclidean SpaceEuclidean 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.

Three Dimensional PlaneThree Dimensional Plane

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.

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.

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 Spherical Coordinates (r, θ, Φ)

Sherical Coordinates Sherical Coordinates (r, θ, Φ)

Set of points where Set of points where ρ ρ (rho) is (rho) is constantconstant

Points where Points where Φ Φ (phi) is constant(phi) is constant

Points where Points where θ θ (theta) is constant(theta) is constant

What happens when they 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 LinesSkew 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.

Example of skew linesExample of skew lines

Four Skew LinesFour Skew Lines

Everyday example of Skew linesEveryday example of Skew lines

Skew lines in Air ShowsSkew lines in Air Shows

Air displaysAir displays

Graphs in 2-DGraphs in 2-D

Surfaces in 3 - DSurfaces in 3 - D

3-D surface3-D surface

Example of function of two Example of function of two variablevariable

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.

FF((x,yx,y) = ) = yy + 6 sin + 6 sin xx + 5 + 5yy22

To find Partial DerivativesTo 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 FGraph of F((x,yx,y) = ) = yy + 6 sin + 6 sin xx + 5 + 5yy22

taking y constanttaking y constant

We see a sine curve at the bottom and this comes from the 6 sin x part of our functionFF((x,yx,y) = ) = yy + 6 sin + 6 sin xx + 5 + 5yy22. . 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 Partial Differentiation with respect to yrespect 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 FGraph of F((x,yx,y) = ) = yy + 6 sin + 6 sin xx + 5 + 5yy22

taking x constanttaking x constant

ParabolaParabola

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.

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