Post on 17-May-2015
description
Three Dimensional Space
Dr. Farhana ShaheenYUC-KSA
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 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 Plane
3-D
3-D
3-D Illusions
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 (r, θ, Φ)
Sherical 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 they are all constant• (one by one)• (ρ, θ, Φ) = (Rrho, Pphi, Ttheta)