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Dear Internet Archive Supporter: Time is Running Out! I ask only once a year: please help the Internet Archive today. We’re an independent, non-profit website that the entire world depends on. Our work is powered by donations averaging about $41. If everyone chips in $5, we can keep this going for free. For the cost of a used paperback, we can share a book online forever. When I started this, people called me crazy.

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Dear Internet Archive Supporter: Time is Running Out! I ask only once a year: please help the Internet Archive today. We’re an independent, non-profit website that the entire world depends on.

Our work is powered by donations averaging about $41. If everyone chips in $5, we can keep this going for free. For the cost of a used paperback, we can share a book online forever. When I started this, people called me crazy. Collect web pages?

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R – of the sphere A sphere (from — sphaira, 'globe, ball' ) is a perfectly round object in that is the surface of a completely round, (viz., analogous to a circular object in two dimensions). Like a, which geometrically is an object in two-dimensional space, a sphere is defined mathematically as the that are all at the same distance r from a given point, but in three-dimensional space.

This distance r is the of the ball, and the given point is the of the mathematical ball. These are also referred to as the radius and center of the sphere, respectively, the longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a of the (sphere) ball. While outside mathematics the terms 'sphere' and 'ball' are sometimes used interchangeably, in a distinction is made between the sphere (a two-dimensional in three-dimensional ) and the ball (a three-dimensional shape that includes the sphere as well as everything inside the sphere). This distinction has not always been maintained and there are mathematical references, especially older ones, that talk about a sphere as a solid, this is analogous to the situation in the, where the terms 'circle' and are confounded. Main article: The basic elements of are and, on the sphere, points are defined in the usual sense. The analogue of the 'line' is the, which is a; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by shows that the shortest path between two points lying on the sphere is the shorter segment of the that includes the points.

Many theorems from hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's, including the. In, are defined between great circles. Spherical trigonometry differs from ordinary in many respects, for example, the sum of the interior angles of a always exceeds 180 degrees. Also, any two spherical triangles are congruent. Eleven properties of the sphere [ ]. A normal vector to a sphere, a normal plane and its normal section.

The curvature of the curve of intersection is the sectional curvature, for the sphere each normal section through a given point will be a circle of the same radius, the radius of the sphere. This means that every point on the sphere will be an umbilical point. Free Download Program Boys Life Howard Korder Pdf Viewer. In their book Geometry and the Imagination and describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the, which can be thought of as a sphere with infinite radius, these properties are: • The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant. The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar of for the, this second part also holds for the.

• The contours and plane sections of the sphere are circles. This property defines the sphere uniquely. • The sphere has constant width and constant girth. The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the, the girth of a surface is the of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.

• All points of a sphere are umbilics. At any point on a surface a is at right angles to the surface because the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a normal section, and the curvature of this curve is the normal curvature, for most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the. Any closed surface will have at least four points called, at an umbilic all the sectional curvatures are equal; in particular the are equal.

Umbilical points can be thought of as the points where the surface is closely approximated by a sphere. For the sphere the curvatures of all normal sections are equal, so every point is an umbilic.

The sphere and plane are the only surfaces with this property. • The sphere does not have a surface of centers. For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures are called the focal points, and the set of all such centers forms the. For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special: • For one sheet forms a curve and the other sheet is a surface • For, cylinders, and both sheets form curves.

• For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere.

• All geodesics of the sphere are closed curves. Are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane, for the sphere the geodesics are great circles. Many other surfaces share this property. • Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume. It follows from.

These properties define the sphere uniquely and can be seen in: a soap bubble will enclose a fixed volume, and minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). • The sphere has the smallest total mean curvature among all convex solids with a given surface area. The is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.

• The sphere has constant mean curvature. The sphere is the only imbedded surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as have constant mean curvature. • The sphere has constant positive Gaussian curvature. Is the product of the two principal curvatures. It is an intrinsic property that can be determined by measuring length and angles and is independent of how the surface is in space. Hence, bending a surface will not alter the Gaussian curvature, and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it.

All these other surfaces would have boundaries, and the sphere is the only surface that lacks a boundary with constant, positive Gaussian curvature, the is an example of a surface with constant negative Gaussian curvature. • The sphere is transformed into itself by a three-parameter family of rigid motions. Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see ). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the. The plane is the only other surface with a three-parameter family of transformations (translations along the x- and y-axes and rotations around the origin).

Circular cylinders are the only surfaces with two-parameter families of rigid motions and the and are the only surfaces with a one-parameter family. See also [ ].

•, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus • ^, p. 223 • ^ Pages 141, 149. Collins Dictionary of Mathematics.. 221 • r is being considered as a variable in this computation • •, p. •; Cohn-Vossen, Stephan (1952).

Geometry and the Imagination (2nd ed.). CS1 maint: Multiple names: authors list () References [ ] has the text of the article. • Albert, Abraham Adrian (2016) [1949], Solid Analytic Geometry, Dover, • Dunham, William.

The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems and Personalities. • Steinhaus, H. (1969), Mathematical Snapshots (Third American ed.), Oxford University Press • Woods, Frederick S. (1961) [1922], Higher Geometry / An Introduction to Advanced Methods in Analytic Geometry, Dover External links [ ].