## AlexandervonHumboldt2 one year ago Sphere Tutorial

1. AlexandervonHumboldt2

Sphere is the locus of points in space equidistant from a given point (the sphere center). The sphere is a surface of rotation formed by the rotation of the semicircle around its diameter. The area of ​​the sphere in degrees taking into account the variability of the dimension values ​​of the arcs of 41,252.96 square meters. degrees. The sphere is a special case of an ellipsoid, which all three axis (the axis, radius) are equal. The sphere is the surface of a sphere. The sphere has the smallest footprint of all the surfaces bounding the volume, and from all surfaces of the area limits the scope of the largest volume. Therefore, the spherical shape of the body found in nature, for example, small droplets of water in free fall acquire a spherical shape because of minimizing the surface area of ​​the surface tension. The volume of the cylinder, the volume inscribed in a sphere tangent to both its base and scope of the cone with apex at the center of one of the cylinder and a base coinciding with the other base of the cylinder, are in the ratio 3: 2: 1. The perfection of spherical shape has long attracted the attention of thinkers and scientists, who with the help of the harmony of the spheres have tried to explain the world. The ancient Greeks emerged of the revolving crystal sphere, to which are attached the stars. Also Wednesday, Greek scientists were cosmological model with a spherical Earth and is attached to a rotating sphere of ether planets. Perceptions of rotating celestial spheres dominated, at least to the Middle Ages and even entered a heliocentric system Copernicus, Common formulas are: The surface area of the sphere: $S=4\pi r^2=\pi d^2$ The volume of the ball bounded by the sphere: $V=\frac{ 4 }{ 3 }*\pi r^3$ Area of segment of a sphere: $S=2\pi rH$ The equation of the sphere in the standart coordinate system: $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2$ Parametric equation of a sphere centered at $x_0, y_0, z_0$. $x=x_0+R*\sin \theta * \cos \phi$ $y=y_0+R*\sin \theta * \sin \phi$ $z=z_0+R * \cos \theta$ The circle lies on the sphere whose center coincides with the center of the sphere, called a great large circle sector. Large circles are geodesics on the sphere; any two of them intersect at two points. In other words, the great circle of peers are lines in the plane, the distance between the points on the field - the length of the arc passing through them the big circle. A corner is between the lines on the plane corresponds to the dihedral angle between the planes of great circles. Many theorems of geometry on the plane and hold in spherical geometry, there are analogs of the theorem of sines, cosines theorem for spherical triangles. At the same time, there are many differences, for example, in a spherical triangle angle sum is always greater than 180 degrees to the three grounds of equality of triangles added their equality on three corners at the spherical triangle can be two or even three right angles - for example, a spherical triangle formed the equator and longitudes 0 ° and 90 °. If we are given the spherical coordinates of the two points, the distance between them can be found as follows: $L=R*\arccos(\cos \theta_1 * \cos \theta_2 + \sin \theta_1 * \sin \theta_2 * \cos(\phi_1 - \phi_2))$ However, if the angle of $\ theta$ does not set between the Z axis and the vector on the point of the sphere, and between this vector and the plane XY , the formula would be: $L=R=\arccos(\sin \theta_1*\sin \theta _2 + \cos \theta_1 * \cos \theta _2*\cos(\phi_1-\phi_2))$ In general, the equation (n-1) -dimensional sphere (in n-dimensional Euclidean space) is as follows: $\sum_{i=1}^{n}(x_i - a_i)^2=r^2$ where $a_1, a_2$ - center of the sphere, and $r$ - radius. The intersection of the two n-sphere is an n-1-dimensional sphere that lies on the radical hyperplane these areas. The n-dimensional space are mutually touch each other (in different parts) of not more than n + 1 fields. n-dimensional inversion takes n-1-dimensional sphere in the n-1-dimensional sphere or a hyperplane.

2. imqwerty

@pooja195 @ItsKoreyyy @DrummerGirl3 @Adi3 @chainedecho @DROPD33DM9

what in the world is that?????!!!!!!!!!

4. DrummerGirl3

Its a sphere tutorial

hates of to @AlexandervonHumboldt2 for writing that much

6. DrummerGirl3

A locus is a set of points that meet a given condition. The definition of a circle locus of points a given distance from a given point in a 2-dimensional plane. The given distance is the radius and the given point is the center of the circle. In 3-dimensions (space), we would define a sphere as the set of points in space a given distance from a given point.

7. Michele_Laino

great job!

8. AlexandervonHumboldt2

thx $$\Huge\color{black}{☺☻☺☻☺☻☺☻☺☻☺☻☺☻☺☻}$$