Equations in R3
In analytic geometry Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry, a sphere with center (x0, y0, z0) and radius r is the locus In mathematics, a locus is a collection of points which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve. For example, in two-dimensional space a line is the locus of points equidistant from two fixed points or from two parallel lines of all points (x, y, z) such that
The points on the sphere with radius r can be parametrized via
(see also trigonometric functions In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications and spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance from a fixed origin, the elevation angle of that point from a fixed plane, and the azimuth angle of its orthogonal projection on that plane, from a fixed direction on the).
A sphere of any radius centered at zero is an integral surface of the following differential form In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. A differential form of degree k, or k-form, on a smooth manifold M is a smooth section of the kth exterior power of the cotangent bundle of M. The set of all k-forms on M is a:
This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek ὀρθός , meaning "straight", and γωνία (gonia), meaning "angle". For example, a subway and the street above, although they do not physically intersect, are orthogonal if they cross at a to each other.
The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension Surface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid locally minimizes surface area. The surface area in relation to the mass of a sphere is called the specific surface area It is a derived scientific value that can be used to determine the type and properties of a material . It is defined either by surface area divided by mass (with units of m²/kg), or surface area divided by the volume (units of m²/m³ or m-1). From the above stated equations it can be expressed as follows:
A sphere can also be defined as the surface formed by rotating a circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius about any diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle. The word "diameter" derives from Greek διάμετρος , "diagonal of a circle", from δια- (dia-), "across, through&. If the circle is replaced by an ellipse In mathematics, an ellipse is the finite or bounded case of a conic section, the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane (the other two cases being the parabola and the hyperbola). It is also the locus of all points of the plane whose distances to two fixed points add to the same, and rotated about the major axis, the shape becomes a prolate spheroid A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters, rotated about the minor axis, an oblate spheroid.
<<Table of Contents A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The | Next>> | Show All>>
