Project #2 - Conic Sections
Conic sections are the various gemetric figures created by the interection of a plane. They are among the oldest curves in history and is one of the oldest area of study for mathmaticians. conics were discovered by Menaechmus (c. 375 - 325 BC), a Greek pupil of Plato and Exodus. He was trying to solve the famous problem duplicating a cube. Euclid studied them and Appollonius reinforced and expanded previous results of conics into a book he named Conic Sections. It is a series of eight books with 487 propositions. He applied his findings to the study of planetary motion and it was used to advance the development of Greek astronomy. It is because of Appollonius that the name ellipse, parabola, and hyperbole were given to conics. Conics evolved even further during the Renaissance with Kepler’s law of planetary motion, Descarte on his work Geometry and Fermat’s coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, and Pascal. We can see conics in satellite dishes, sharpening pencils, automobile headlights, when a baseball is hit, telescopes, and much more. Physicians apply conics in treating kidney stones. Even, John Quincy Adams used conics to eaves drop on members of the house of representatives from his desk in the U.S. Capitol building.
Conic Sections are the improved curves produced by the intersection of a plane with a cone. For a plane perpendicular to the axis of the cone, a circle is produced. The definition of a cone includes the surface generated by a straight line that moves so that it always intersects the circumfrence of a given circle and passes through a given point not on the plane of the circle. The point, called the vertex of the cone, divides th...
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...t a polar equation for the ellipse. In defining the quality , our polar equation then becomes
;
since , we have  and thus

Because a , b , and f must be positive lengths, the quantity e must be positive; it is possible, though, for a to equal b (in which case, the ellipse becomes a circle), so that f = 0 , so e may also be zero. Since r , the distance from the focus at the origin to a point on the ellipse, must also be positive, we require that . No point on the ellipse is at the origin, so r ≠ 0 ⇒ e ≠ 1 . For an ellipse, then, 0 ≤ e < 1 .
This unification of the conic sections under one expression simplifies the proof that an object, subject to a force which varies with the inverse square of the distance from the agent of that force (such as gravity or the electric force of attraction or repulsion), will follow a path which is one of the conic sections.
2.6 Why is the line curved rather than straight? What kind of distance is being computed here?
Aristotle, R. P. Hardie, and R. K. Gaye. Physics. Adelaide: The University of Adelaide Library, 2000. Print.
3. I. Boldea and S.A. Nasar, Linear Motion and Electromagnetic Systems. John Wiley & Sons Inc., 1995.
...does increase, proving the fact that the parabola not only become more definite in shape as the distance increases, but so does the trajectory and height.
He specifically used parabolas in many of his paintings and drawings. In paintings such as the Mona Lisa, The Virgin of the Rocks, Child With Saint Anne, Lady With An Ermine, and many others, parabolas are seen. The parabolas in these art works can be seen often or scarcely depending on the painting or drawing. Parabolas are commonly seen on faces or body parts. In the Mona Lisa painting, the most noticeable feature of this painting can be the slight smile on the face. You can clearly see the parabola if you look closely. In the painting, The Virgin of the Rocks, the use of parabolas is very common. You can see parabolas were by Leonardo da Vinci to shape the faces of the people, to make the eyebrows, and lips. In the cut of the clothing of the Virgin, you can see a parabola as well. The other woman in the painting has a cloth hanging on her shoulder that is shaped in a parabolic form. Looking at the shape of the arm of the baby on the left is even shaped in a parabolic manner. Parabolas are also seen all throughout the background of this painting. You can see them on the rocks or pillar-like shapes. On the painting, Child With Saint Anne, parabolas can be seen on the faces to make the eyebrows on the woman in the back. This is also the case in her smile or grin. Throughout the clothing of the women, the use of parabolas are common. If you look closely at the knees of the women, you can see that Leonardo da Vinci used parabolas as a guide to paint and/or draw them. The painting, Lady With An Ermine, has many visible parabolas. The more obvious parabolas are on the neck and chest of the lady. Her necklace forms two perfect parabolas, one above the other. The general shape of the lady from her shoulders down is a wide parabola. Similar to many other da Vinci paintings, the face contains many parabolas. The eyebrows, chin, and hairline were clearly used by parabolas to make. As we
Hesiod, writing in the 8th century BC, used celestial bodies to indicate agricultural cycles: "When the Pleiads, Atlas' daughters, start to rise, begin your harvest; plough when they go down" (Hesiod 71). Later Greek scientists, such as Archimedes, developed complicated models of the heavens-celestial spheres-that illustrated the "wandering" of the sun, the moon, and the planets against the fixed position of the stars. Shortly after Archimedes, Ctesibus created the Clepsydra in the 2nd century BC. A more elaborate version of the common water clock, the Clepsydra was quite popular in ancient Greece. However, the development of stereography by Hipparchos in 150 BC.
In the parasternal short-axis view, PDA flow is usually detected along the left lateral wall of the main pulmonary artery, and is usually directed towards the transducer. Cranial tilting of the transducer demonstrates the PDA. By sliding the transducer superiorly into a high left parasternal window and clockwise rotation, the pulmonary artery (PA) bifurcation can be seen. In this view, the LPA goes leftward of the descending thoracic aorta toward the left scapula. From this view of the branch pulmonary arteries, counterclockwise rotation of the transducer toward 12 o’clock demonstrates the long-axis of the PDA, which is located between the LPA and the descending aorta.
Now if we reverse our variables and look instead at the values of (m, c) as a function of the image poin...
Believed to have been a pupil of Pythagoras, Alcmaeon pioneered anatomical dissection, a field in which Galen will later innovate. Through dissection Alcmaeon came to believe that the sensory organs were connected to the brain by channels. While he did not initially perceive the nervous system he is given credit for identifying the optic nerve. His Pythagorean influence in evidenced in his view of natural opposing powers, e.g. the wet, dry, the hot, the cold, the sweet, the bitter etc. The concept of symmetry and balance are part of Pythagorean philosophy as well as with artists in ancient Greece who’s sculpted works with the purpose of depicting excellence through natural symmetry in the male form. For Alcmaeon, proper health was achieved and maintained so long as the opposing powers remained in balance. Alcmaeon may also have been the first to identify “environmental problems, poor nutrition” and general “lifestyle” issues as factors related to illness rather than divine explanation (Nordqvist,
Some historians believe she may have ever surpassed her father’s knowledge. One important impact that Hypatia had on math, was edition the on the Conics of Apollonius. The Conics of Apollonius divided cones into different parts by a plane. This concept was extremely complex, and developed the ideas of parabolas, hyperbolas, and ellipses. Though Hypatia did not originate this concept, “ With Hypatia's work on this important book, she made the concepts easier to understand, thus making the work survive through many centuries”
points and leading at once to the Cartesian equation of the evolute of any conic.
increases in the opposite or negative direction until it attains maximum negative value at 270 degrees, and finally decreases to zero value again at 360 degrees. It follows, then, that the induced emf can be completely described by the relation.
Mathematicians, engineers and scientists encounter numerous functions in their work: polynomials, trigonometric and hyperbolic functions amongst them. However, throughout the history of science one group of functions, the conics, arise time and time again not only in the development of mathematical theory but also in practical applications. The conics were first studied by the Greek mathematician Apollonius more than 200 years BC. Essentially, the conics form that class of curves which are obtained when a double cone is intersected by a plane. There are three main types: the ellipse , the parabola and the hyperbola . From the ellipse we obtain the circle as a special case, and from the hyperbola we obtain the rectangular hyperbola as a special case. These curves are illustrated in the following figures. cone-axis
O’Connor, J. J. and Robertson, E.F. “Sir Isaac Newton.” Mac Tutor History of Mathematics, Inc. Jan 2000. Web. Aug 31 2011.
As what can be seen in the data part of the report, the given values for x and y can be interpreted as a graph. With this graph, it can be interpreted to describe the phenomenon that is taking place. This is how graphs and equations can help the study of science understand how and why things are taking place.