Supplementary mathematics/Spatial geometry

Spatial geometry refers to Euclidean geometry in three-dimensional space. A space where height exists apart from length and width. Spatial geometry requires a lot of imagination. The whole world around us is three-dimensional and spatial. Any volume you know should have its properties calculated in the subject of spatial geometry. Shapes such as spheres, cones and cylinders are of this category. Spatial geometry includes three-dimensional spatial items (length, width, height).

Such as: area, volume, geometric volumes, polyhedra, conic section, three-dimensional space, spherical geometry, spherical coordinates, cylindrical coordinates and...

History

The history of spatial geometry dates back to ancient Greece, the Pythagoreans dealt with regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus measured them and proved that the pyramid and the cone are one-third the volume of a prism and a cylinder on the same base and the same height. He probably also discovered the proof that the volume enclosed by a sphere is proportional to the cube of its radius.

Definition of topics

area and volume

Volume: The amount of space occupied by an object is called volume. The volume unit is equal to the cubic unit. Volume is a quantity of three-dimensional space that is limited by a specific boundary, for example, it is the space occupied by a substance (solid, gas, liquid, plasma) or its shape. Volume is a sub-unit of SI, which is the unit It is meter to the power of 3 (cubic meter). The volume of a container is equal to the volume of the liquid that fills it. To calculate the volume of certain 3D shapes, there are specific relationships that are simple relationships for simple shapes with geometric regularity. For complex shapes that do not have a simple relationship to calculate the volume, the volume can be obtained from integral methods. The volume of one-dimensional shapes, such as a line, or two-dimensional shapes, such as a plane, is zero.

Area: It is a type of quantity that calculates the surface area of three-dimensional objects and the internal value of two-dimensional objects. The area unit is equal to the square unit. Area is a quantity that expresses the extent of an area on a plane or on a curved surface. slow The area of the plane region or "plane area" refers to the area of a planar layer or layer, while "surface area" refers to the area of an open surface or boundary of a three-dimensional object. Area can be understood as the amount of material of a given thickness required to form a model of a shape, or the amount of paint required to cover a surface with a layer. This two-dimensional analog is the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

Geometric volumes and non-geometric volumes

Non-geometric volumes are complex volumes whose volumes are hard to obtain. But their area can be obtained, but it is a bit complicated. To obtain non-geometric volumes, we first pour water in a beaker. After we fill it with water and measure the amount of liters, we drop the non-geometric object into water with this The water rises, then we subtract the amount of water that has risen with a non-geometric volume by the amount of water that was determined before, and then we measure and write its volume.

"Geometric volumes" = geometric volumes are objects for which we can write surface and volume formulas. We can find the volume of those geometric objects by pattern-finding method by analyzing and measuring the volume of the corresponding components. And by summarizing and formulating it, we can get the formula of its volume. To find its area, we first calculate the area of its components by analyzing and drawing the shape in a continuous and discrete way, and write its formula by analysis.

Example = sphere, pyramid, prism, polyhedron, cylinder, cone and cube, tetrahedron, parallelogram

definition of prism, sphere, pyramid, polyhedron

"Definition of Prism": A prism is a volume that has two bases, lateral faces, vertices and edges. The faces of a prism are rectangular and the number of its faces is equal to the number of sides of its base, the number of its vertices is twice as many as the faces and The number of edges is three times the face of the prism. The faces of the pyramid are obtained by the formula n+2, because the number of faces of the prism hub is always two more than the side face, because the other two faces are the base of the prism. In geometry, a prism is a polyhedron with a base of n- side, transferred base polygon (in another plane) and n other faces which are necessarily all parallelograms and connect the corresponding vertices of two n-gons. All cross sections parallel to the base are the same. Prisms are named according to the number of their base sides; So, for example, a prism with a pentagonal base is called a pentagonal prism. The definition of a prism to a pyramid is that a prism is the same as a pyramid, but its apex is at infinity.

Definition of a pyramid: A pyramid is a volume whose faces intersect at a point and whose faces are triangular with a base. The number of edges of the pyramid is twice the number of the sides of the base. In fact, a pyramid is a three-dimensional shape that is formed by connecting a point in space to all closed points on the plane. That point is called the top of the pyramid and that flat shape is called the base of the pyramid. The base of the pyramid is an arbitrary polygon and the other faces are equilateral triangles that connect to each other at the vertex. The vertical line that connects the vertex to the base is called the height of the pyramid. Among the most famous structures in the world in the form of a pyramid, we can mention the Egyptian triple pyramids.

"Definition of a sphere": A sphere is a perfectly round geometric object in three-dimensional space. For example, a ball is a sphere. A sphere, like a circle in two dimensions, is perfectly symmetrical around a point in three-dimensional space. All points on the surface of the sphere are at the same distance from the center of the sphere. The distance of these points from the center of the sphere is called the radius of the sphere and is represented by the letter "r". The longest distance from both sides of the sphere (which passes through the sphere) is called the diameter of the sphere. The diameter of the sphere also passes through its center and therefore its size is twice the radius. A sphere is a set of points in space that has a circular base and radius, which is a regular polyhedron. The sphere is the result of the period of a semicircle and a circle around the diameter, which rotates 180 degrees in a circle and 360 degrees in a semicircle. We divide the faces of the sphere into several degrees based on the division of its area, which is 360 degrees.

Polyhedron Definition: A polyhedron is a solid geometric object in three-dimensional space that has smooth faces (each face in one plane) and sides or edges located on a straight line. So far, no single definition has been provided for it. A tetrahedron is a type of pyramid and a cube is an example of a hexagon. A polyhedron can be convex or non-convex. Polyhedrons such as pyramids and prisms can be made by extruding two-dimensional polygons. There can only be a finite number of convex polyhedra with regular faces and equiangular shapes, including Platonic solids and Archimedean solids. Some Archimedean solids can be made by cutting the top pyramid of Platonic solids. Due to the simplicity of construction, polyhedra are used in most architectural works such as geodesic domes and pyramids. Recently, due to the use of shapes, interest in multifaceted surfaces has increased. Some compact molecules and atoms, especially crystalline structures and Platonic hydrocarbons, as well as some radials have a shape similar to Platonic solids. Platonic solids are also used in making dice. Polyhedra have different characteristics and types and are placed in different symmetry groups. Other polyhedra can be created by operations on any polyhedra. Some of them have relationships with each other. Polyhedra have been of interest since the Stone Age. The sphere is also considered as a family of polyhedra. Cube, tetrahedron, parallelogram are geometric volumes that are also considered polyhedra.

conic section

In mathematics, a conic section (or simply a conic, sometimes called a quadratic curve) is a curve obtained as the intersection of the surface of a cone with a plane. Three types of conic sections are hyperbola, parabola and ellipse. The circle is a special case of the ellipse, although historically it is sometimes called the fourth type. Ancient Greek mathematicians studied conic sections, culminating in Apollonius Perga's systematic work on their properties around 200 BC.

3D space

In mathematics, "3D space" is a vector space with three dimensions and a geometric model of the physical world in which we live. The three dimensions are commonly known as length, width, and height (or depth), although this naming is optional.

Spherical geometry

Spherical geometry is the branch of geometry that deals with the two-dimensional surface of a sphere. This is an example of geometry unrelated to Euclidean geometry. The practical application of spherical geometry is in the field of aviation and astronomy. In Euclidean geometry, straight lines and points are the main concepts. In Korea, dots are defined in their usual meaning. In Euclidean geometry, lines do not mean a straight line, but in the concept of the shortest distance between two points, a straight line is proposed, which is called a geodesic. On a sphere, geodesics are great circles. Other geometric concepts are defined on the page, except that a straight line is used instead of a great circle. Therefore, in spherical geometry, angles are defined between great circles, and as a result, spherical trigonometry is different from ordinary trigonometry in many ways. For example: the sum of the internal angles of a triangle is more than 180 degrees. Spherical geometry is not elliptic (Riemannian) geometry, but this feature that a line from a point cannot have a line parallel to it is common to both. In isometrics of spherical geometry with Euclidean geometry, the line from a point has a line parallel to itself, and in isometry with hyperbolic geometry, the line from a point has two lines parallel to itself and infinity. Concepts of spherical geometry may be applied to the spindle sphere, although slight modifications must be made to certain formulas.

spherical coordinates

In mathematics, spherical coordinates are for three-dimensional space, in which the position of a point is determined by three numbers: the "radial distance" of that point from a fixed origin, "its polar angle measured from a direction," the apex fixed, and its orthogonal "orthogonal" angle on a reference plane that passes through the origin and is perpendicular to the vertex, is measured from a fixed reference direction in that plane. It can be seen as a three-dimensional version of the polar coordinate system.

The use of symbols and the order of coordinates are different in sources and disciplines. This paper uses the ISO convention often encountered in physics: it shows the radial distance, the polar angle, and the azimuth angle. In many math books, the radial distance shows the azimuthal angle and the polar angle and changes the meanings of "θ" and "φ". Other conventions are used, such as ``r for the radius from the ``z axis, so great care must be taken to check the meaning of the symbols.

According to the conventions of geographic coordinate systems, positions are measured by longitude and latitude and altitude (elevation). There are a number of celestial coordinate systems based on different base planes and with different terminology for different coordinates. Spherical coordinate systems used in mathematics usually use radians instead of degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0 degrees) to east. (90 degrees) like the horizontal coordinate system. . The polar angle is often replaced by the elevation angle measured from the reference plane, so that the zero elevation angle is at the horizon.

The spherical coordinate system is a generalization of the two-dimensional polar coordinate system. It can also be extended to higher dimensional spaces and then it is called a hyperspherical coordinate system.

cylindrical coordinates

Cylindrical coordinate is a type of orthogonal coordinate in which a point is considered in space on the base of a cylinder. The location of that point is expressed based on the radius and height of the cylinder (r and z) and the angle that the radius of the base passing through that point makes with the x axis (θ). This device, in two-dimensional mode, is converted to polar coordinates by removing z. In physics and especially in the topics of electromagnetics and telecommunications, instead of r, θ, z, the letters ρ, φ, z are used respectively.