# Analytical geometry

The basic concept of analytical geometry and its applications

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### Analytical geometry

• 2. INRODUCTION Analytic geometry can be defined as a branch of mathematics that is concerned with carrying out geometric investigations using a various algebraic procedures. It is also known as coordinate geometry it is a study of geometry using a coordinate system. It is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes, numerical definition from shapes, numerical definition and representations
• 3. Rene Descartes HISTORY French mathematician and philosopher, Rene Descartes is generally credited for the invention of the ideology behind analytical geometry. In the article Oiscoues de la method (1639) Descartes outlined the principles behind analytical geometry. Analytical geometry grew out of need for established uniform techniques for solving geometrical problems, the aim being to apply them to the study of curves, which are of particular importance in practical problems. The aim was achieved in the coordinate method
• 5. coordinate In analytic geometry, the plane is given a coordinate system. By which every point has a pair of real number coordinates. Similarly, Euclidean space is given a coordinate where every point has three coordinates. The value is the coordinates depends on the choice of the initial point of origin
• 6. Types 1. Cartesian coordinates 2. Polar coordinates 3. Cylindrical coordinates 4. Spherical coordinates
• 7. The Distance formula e distance between points P(x1,y1) and Q(x2,y2) on the coordinate plane can be found by the formula below d = √(x2-x1)2 + (y2 -y1)2 Midpoint formula In analytical geometry, P&Q are specified by coordinates, and so the midpoint formula identifies point M by a pair of coordinate as well M (x,y) = (x1+x2/2 , y1+y2/2) Slope The slope of a line is a measure of how tilted the line is, Aline of slope m =0 is completely horizontal. A line tilting upwards is positive slope.(m>0) while a line tilting downwards is negative slope (m<0) A slope of a vertical line is m =y2-y1 / x2-x1
• 8. Conic sections In the Cartesian coordinate system, the graph of a quadratic equation in two variable is always a conic section
• 10. The cartesian plane is determined by two perpendicular number lines called the x-axis and the y-axis. These axes, together with their point of intersection(the origin), allow you to develop a two- dimensional coordinate system for identifying points in a plane. To identify a point in space, you must introduce a third dimension to the model. The geometry of this three-dimensional model is called solid analytic geometry. The three dimensional coordinate system
• 13. Analytical geometry is applied theoring mathematics, trigonometry, algebra, spectronomy, computational arithmetic Parallel equation is used to demonstrate graph statistics in analytical computation An application of analytical geometry would be used to demonstrate projection in a constant ex. one can access the directive in principle structure by using the hyperbola The three dimensional coordinate system issued in chemistry to help understand the structure of crystals. ex, isometric crystals are shaped like cubes The three dimensional coordinate system can be used to graph equations that model surfaces in space such as the spherical shape of the earth
• 14. IMPORTANCE OF ANALYTICAL GEOMETRY  The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra, and vice versa. We can visualize the numbers as points on a graph, equation as geometric figures and geometric figures as equation.  Analytical geometry was originally formulated in order to be able to make effectively investigations on plane geometry but the concept of analytical geometry can also be used to explore other spaces of higher dimensions.
• 18.  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 contours and plane sections of the sphere are circles.  The sphere has constant width and constant girth.  All points of a sphere are umbilics.  The sphere does not have a surface of centers.  All geodesics of the sphere are closed curves. ELEVEN PROPERTY OF SPHERE
• 19. Of, all 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.  The sphere has the smallest total mean curvature among all convex solids with a given surface area.  The sphere has constant mean curvature.  The sphere has constant positive gaussian curvature.  The sphere is transformed into itself by a three-parameter family of rigid motions.
• 21. PLANET Planets are round because of its Gravitational field. As a planet gets massive enough, internals heating takes over and the planet behaves like a fluid. Gravity then pulls of all of the material towards the center of mass. Because all points on the surface of a sphere are an equal distance from the center of mass, planets eventually settle on a spherical shape. For, major planets, one of the requirements is that it’s large enough for it’s gravity to pull it into a sphere.
• 23. Example A spherical fish tank has a radius of 8 inches. Assuming the entire tank could be fill with water what would the volume of the tank be?
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