Conic Sections
The term conic sections is used when discussing the derivation of a line that is a locus of points equal distance from either a line, a point, both a line and a point, two lines, etc. The term conic sections also can be used when discussing certain planes that are formed when they are intersected with a right circular cone. The planes, or lines as we know them, consist of the circle, the ellipse, the parabola, and the hyperbola. (West, 112) There are different ways to derive each separate curve, and many uses for them to be applied to as well. All of which are an important aspect to conic sections.
The cone is a shape that is formed when you have a straight line and a circle, and the straight line is moved around the circumference of the circle while also always passing through a fixed point at a distance away from the circle. The parts formed are labeled the upper nappe, the lower nappe, and the vertex, (Prime, 1) as described in the diagram below in diagram 1:
The cone is then used with the help of a right plane to form the different circles, parabolas, ellipses, and hyperbolas, as shown below in diagram 2 on the next page. Taking a flat plane that would be parallel to the base of the cone, and intercepting it with a single nappe of the cone produces the circle. The ellipse is formed by the intersection of the cone with a flat plane that intercepts one nappe of the cone, but is not parallel to the base, and is not parallel to any other side of the cone. The parabola is a curve produced by the intersection of the cone with a flat plane of a single nappe, parallel to the remaining slanted side. Finally, the hyperbola is produced when you take a flat plane and intercept it with the cone so that it pa...
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...er way that the hyperbola is applied to everyday science is in a popular radar system known as LORAN. This system uses foci and hyperbolic curves to determine the location of ships and other such objects.
As one can see, there are many formulas that can be derived and applied when using conic sections. Their understanding and comprehension is important for laws pertaining to astronomy, and many other applications as previously mentioned.
Mr. Finta, I would just like to let you know that in the original version of this report there was 14 and a half pages with the vertex equations thoroughly covered and with a great proofreading job. The essay was lost on my computer and I had to start from what I last saved it at, 4 pages. I re-built it as best I could but it is nowhere near the quality of its original counterpart as this one was rushed to be completed.
...t the very end. In the case of The Cone it has a very detailed
If the cone is made up entirely of rock debris, we found very high gradient for the space occupied by the base. Here we find the gradient of 30 degrees and sometimes up to 40 degrees Celsius and these forms usually arise as a result of volcanic eruptions and represented in the islands in Indonesia.
He created the height and distance in geometry. He also invented another theorem called the intercept theorem. Thales intercept theorem states that DE = AE = AD
Before the age of algebra and analytic Geometry there was Pythagoras: Philosopher, well-known Mathematician, scientist, and a religious teacher. In life this famous Greek thinker developed and coined Pythagoreanism, his own branch of education which would set his theorems to become quite important in geometry. Though there is not much known about Pythagoras’s personal life, relative to more recent famous mathematicians, his life and contributions to mathematics are important in developing the field of geology into what it is today.
Believe it or not, Geometry is actually useful! All our lives we have been told that we would use this in our lives and we have thought “no we won't.” but we do use it in life. Geometry is used for home decorating. Also architects use it for home building.
The following formula is used to calculate the surface area of the spherical coordinates system.
One of the ways that I use geometry in everyday live is when I play football, because we have to know when we throw the ball, where it is supposed to intercept the receiver. Also in the mornings when you pour a glass of milk and you don’t want to overfill the glass then you have to know the volume of the glass. Also, when you are drawing the blueprints of a house and you need to draw the shapes of rooms and have to have the sizes then that is also geometry. Another thing you use geometry is when you are driving and you are keeping a distance between you and other cars because you know how long it will take you to stop your car before you have a collision. According to teachnology, “For instance, the size or area of a specific appliance or tool
...ture points are selected on these vectors. The number of points on the radial line is called radial resolution and along the angular line is called angular resolution.
Conic sections are used all over the world. Conic sections are used in things such as bridges, roller coasters, stadiums, and other objects. A conic section is the intersection of a plane with a cone. The changes in the angle of the intersection produce a circle, ellipse, parabola, and hyperbola.
I think the great example of using of calculus and analytical geometry in our life is radar and setellite .
Fractal Geometry The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with fractals - a concrete one. Fractals go from being very simple equations on a piece of paper to colorful, extraordinary images, and most of all, offer an explanation to things. The importance of fractal geometry is that it provides an answer, a comprehension, to nature, the world, and the universe.
Figure 1 56 Example on the direction of line sight , calculated depend on the relation between the center of pupil and corneal reflection.
Where CNN is the normal coefficient for the nose, σ, the cone semi-vertex angle is given by
A spheroid is an ellipsoid having two axes of equal length, making it a surface of revolution.