Menaechmus sufficiently advanced the theory of the conic sections so that in antiquity these curves were called the Curves of Menaechmus. The cones were obtained by rotating a right triangle around one of its short sides, and a conic section obtained as section of conic surface by a plane orthogonal to the generating line. The picture from the work quoted bellow shows a rightangle , obtusangle or acutangle conic, as they were called.
Besides Menaechmus, other Greek geometers were studying conic sections and their properties before Apollonius. Some of them were. Apollonius of Perga c. He is famous for his writings on conic sections, and is often wrongly designated as their inventor. His plagiarism? The author of this text states,. On this point, it seems that our commentator [E] has been impressed by another "historian" of the theory of conics, Pappus.
The presentation that he devotes to the treaty of Apollonius begins: "Apollonius has given us eight books on the conics, having completed the four books of the Conics of Euclid, and having added four other books. Aristaeus, author of The Five Books concerning Solid Loci , still available today, following the Conics, had however, as the predecessors of Apollonius, called one of the conical sections the 'acutangle cone section', the another the 'rightangle cone section' and the other still, the 'obtusangle cone section'.
Skipping centuries, I would attract attention to Philippe de la Hire, — who was inspired by Apollonius. An abundant source is this thesis. De la Hire used an early projective method : Every conic section can be obtained from a circle by a projection. This method was developed in later centuries. It is worth noting the approach of Steiner. A nice proof that the focal definition of ellipse sum of distances to two given points, foci, is constant , and ellipse as a section of a conic surface by a plane, is due to Germinal Dandelin and Adolphe Quetelet and is dated to The picture bellow is from wikipedia.
The article Conic sections on Wikipedia is excellent. For the early history of conics in Europe, see the paragraph Europe and the references therein. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? In fact, all orbital motion under the influence of gravity can be described using one of the four conic sections based on the mass and speed of the body in orbit.
Today, the conics are used to describe the motion of myriads of objects from sub-atomic particles to satellites and whole galaxies. Ellipsographs also known as elliptographs are devices used to draw ellipses.
Why would you need to draw an ellipse? These curves arise most often in the areas of architectural and engineering drawing as well as in art and graphic design when drawing in perspective. When drawing plans or blueprints in perspective, a circle when viewed from an angle appears as an ellipse. Many windows, vaulted ceilings, stairs, bridges and arches are elliptical in design and need to be rendered accurately in technical drawings.
An ellipse is an elongated circle with a center C as well as two foci designated by F. The standard equation for an ellipse in Cartesian coordinates is:. The standard description of an ellipse is the set of all points whose distance to the two foci is a constant.
The eccentricity e of an ellipse is a number between 0 and 1 that indicates how elongated the curve is. Thus a circle is a special case of an ellipse where the two foci have come together. The larger e, the more elongated the ellipse is. If you let e equal 1, your ellipse would end up being a line segment, it would have zero width.
The easiest way to draw an ellipse it to take a length of string and tie it to make a loop and place it around two tacks pushed into a piece of paper. With a pencil pull the string taught and start drawing, keeping the sting at the same tension the whole while figure 5.
Since the length of string does not change, the total distance to the two foci remains constant. The curve that results is an ellipse.
You can draw a perfect circle by simply using only one pin. Fig 5. Image of drawing an ellipse using string and pins by the author. There are several mechanical devices that draw an ellipse, but most of them use a method equivalent to this simple string procedure. The simplest ellipsograph used in technical drawing is a template or elliptic curve made of wood or plastic created using a variation on the string method. This type of ellipsograph is static and can only draw one size ellipse but is easy and inexpensive to make.
A draftsman may have a set of several sizes of these templates. Item The most common device for drawing ellipses is by using an elliptical compass or elliptical trammel , often referred to as the Trammel of Archimedes. The focus and directrix of an ellipse were considered by Pappus. Kepler , in , said he believed that the orbit of Mars was oval, then he later discovered that it was an ellipse with the sun at one focus.
In fact Kepler introduced the word "focus" and published his discovery in The eccentricity of the planetary orbits is small i.
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