Spherical Geometry :- The geometrical symmetry is associated with the two dimensional surface of the sphere. It is not Euclidean. The main application of the spherical geometry is in navigation and astronomy. The plane geometry is associated with the point and lines. On the sphere, points are defined as usual sense. The straight lines are not defined as the usual sense. In spherical geometry angles can be defined between great circles, resulting a spherical trigonometry which is differ from the ordinary trigonometry in many respect; the sum of the interior angles of a triangle exceeds one hundred and eighty degrees.
Let r is the radius of the sphere. The volume of the sphere is 4/3 pi r3 cubic meter. To find the volume of the sphere we can divide it into number of infinitesimally small circular disk of the thickness dx. The calculation of the volume of the sphere can be done as below. The surface area of the disk is equal to pi r².
Now the volume of the sphere can be found by finding the integral of the area within the limits of minus r to plus r. The formula can be derived more quickly by finding the surface area and then by integrating it within the limit of zero to r. The spherical geometry is not the elliptical geometry but shares with the geometry the property that a line has no parallel through a given point. The real projected plane is closely associated with the spherical geometry.
Midpoint Formula Geometry : - The point which is exactly at the centre of the two points is known as the middle point. This point divides a given line segment between the two equal halves. The middle point is called the midpoint in the geometry. The midpoint formula has been taken from the section formula. We have to find a point which divides a line in a particular ratio in the section formula.
In the section formula a point which divides the line in the m and n ratio is to find out. If the value of the two ratios which are represented by m and n is equal or m= n then we get the mid points at the given line.
Let AB is a line and P is a point anywhere in between A and B. Let the coordinates of point A are (x₁, y₁) and coordinate of point B are (x₂, y₂). P is the point which lies in between points A and B has the coordinates (x, y). The coordinates on the mid-point of the line segment joining the points (x₁, y₁) and B(x₂, y₂) are given by {(x₁+x₂)/2 ,(y₁+y₂)/2}. these are obtained by replacing l and m in the above formula. If a point P divides the given line segment joining the points (x₁, y₁) and B(x₂, y₂) , then the coordinate of point P are given by {(x₁+kx₂)/(k+1) ,(y₁+ky₂)/(k+1)} These are obtained by dividing the numerator and denominator in the above expression the replacing l/m the by k.
Let r is the radius of the sphere. The volume of the sphere is 4/3 pi r3 cubic meter. To find the volume of the sphere we can divide it into number of infinitesimally small circular disk of the thickness dx. The calculation of the volume of the sphere can be done as below. The surface area of the disk is equal to pi r².
Now the volume of the sphere can be found by finding the integral of the area within the limits of minus r to plus r. The formula can be derived more quickly by finding the surface area and then by integrating it within the limit of zero to r. The spherical geometry is not the elliptical geometry but shares with the geometry the property that a line has no parallel through a given point. The real projected plane is closely associated with the spherical geometry.
Midpoint Formula Geometry : - The point which is exactly at the centre of the two points is known as the middle point. This point divides a given line segment between the two equal halves. The middle point is called the midpoint in the geometry. The midpoint formula has been taken from the section formula. We have to find a point which divides a line in a particular ratio in the section formula.
In the section formula a point which divides the line in the m and n ratio is to find out. If the value of the two ratios which are represented by m and n is equal or m= n then we get the mid points at the given line.
Let AB is a line and P is a point anywhere in between A and B. Let the coordinates of point A are (x₁, y₁) and coordinate of point B are (x₂, y₂). P is the point which lies in between points A and B has the coordinates (x, y). The coordinates on the mid-point of the line segment joining the points (x₁, y₁) and B(x₂, y₂) are given by {(x₁+x₂)/2 ,(y₁+y₂)/2}. these are obtained by replacing l and m in the above formula. If a point P divides the given line segment joining the points (x₁, y₁) and B(x₂, y₂) , then the coordinate of point P are given by {(x₁+kx₂)/(k+1) ,(y₁+ky₂)/(k+1)} These are obtained by dividing the numerator and denominator in the above expression the replacing l/m the by k.
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