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Formule Ellisse Analytical Essay




                                    Tips for Your Analytical Essay


1. Your essay must address and respond to the assignment description.  Most students fail or get low grades because they fail to read the entire assignment, including the grading criteria.

2. Make sure you develop an argumentative analytical essay (i.e., your essay must include an arguable THESIS at the end of your introduction, which you should later develop in the body of your essay through an ANALYSIS of the selected work of art and illustrate with SPECIFIC EVIDENCE).  Consider the following formula to help you develop a working thesis for your essay: “In [title of art piece], the author challenges/reinforces traditional notions of gender/female sexuality/standards of masculinity/etc. by [doing blah, blah, blah].”

3. Your essay must contain INTRODUCTION + BODY + CONCLUSION + WORKS CITED.  Forget about the 5-paragraph essay; those only worked in high school, when the essays were shorter and less complex.

4. All your paragraphs should be fully developed and include transitions.  The paragraphs in the body of your essay should contain a topic sentence introducing the topic to be discussed and relating back to the thesis.

5. Avoid “lab talk” (e.g., “In this paper I will prove…”) and phrases like “I believe that” or “In my opinion.”  Your reader assumes that everything you write that you do not attribute to another author is your opinion.  See Dr. Easton’s handout for more information.

6. Do not abuse plot summaries and/or unnecessary long descriptions.  Remember that your argument is based on an analysis; you’re not writing a book report, but an argument.  Consider including a brief summary of your work of art (in the case of novels, plays, movies, and the like) or a brief description of it (in the case of paintings and sculptures, for instance) in the introduction.  Later, as Celia Easton points out, “Your job is to remind your audience of passages in the text that provide evidence for the argument you want to create about your text, not to describe the plot to someone who has never read the text.”

7. Select lines, quotes, passages, or specific details to discuss to make a claim about the whole work.

8. Make sure your essay follows a logical structure and organization.  It is not necessary to imitate the chronology of the literary work you are analyzing.

9. Avoid generalizations and oversimplifications, such as “all men think…” or “since the beginning of times.”

10. Remember you need to incorporate at least oneacademic (non-fictional) sourceto develop your argument.  Check our website for more information about what counts as an academic source.

11. Don’t let your secondary sources dominate your essay.  In order to avoid this problem, use a yellow marker and highlight every sentence in your essay stating ideas that are not your own (quotes, paraphrases, and summaries of other people’s works).  If you see too much yellow in your paper, chances are your voice and ideas have not been fully develop.

12. Quote only passages that would lose their effectiveness if they were paraphrased.  Never use a quotation to substitute for your own prose.  Always include a tag line on any quotation in order to introduce it (e.g., “According to author X, …” or “As author Y points out, …”)

13. Cite your sources properly in MLA style.  When in doubt, ask.

14. Make sure your essay meets the length requirement: 4-5 pages, including “Works Cited” (at least 4 FULL pages).

15. Read Celia Easton’s “Conventions of Writing Papers about Literature.”

16. Check the links included in the online version of the grading criteria for the assignment.

17. Consider coming to my office hours and/or going to the Writing Center for help with your writing.  Note: I will only address questions about your essays by e-mail only if it takes me a couple of lines to answer.  Don’t e-mail me your drafts.

This article is about the geometric figure. For other uses, see Ellipse (disambiguation).

"Elliptical" redirects here. For the syntactic omission of words, see Ellipsis (linguistics). For the punctuation mark, see Ellipsis.

In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse having both focal points at the same location. The shape of an ellipse (how "elongated" it is) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.

Ellipses are the closed type of conic section: a plane curve resulting from the intersection of a cone by a plane (see figure to the right). Ellipses have many similarities with the other two forms of conic sections: parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder.

Analytically, an ellipse may also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the directrix) is a constant. This ratio is called the eccentricity of the ellipse.

An ellipse may also be defined analytically as the set of points for each of which the sum of its distances to two foci is a fixed number.

Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency. A similar effect leads to elliptical polarization of light in optics.

The name, ἔλλειψις (élleipsis, "omission"), was given by Apollonius of Perga in his Conics, emphasizing the connection of the curve with "application of areas".

Definition of an ellipse as locus of points[edit]

An ellipse can be defined geometrically as a set of points (locus of points) in the Euclidean plane:

  • An ellipse is a set of points, such that for any point of the set, the sum of the distances to two fixed points , , the foci, is constant, usually denoted by In order to omit the special case of a line segment, one assumes More formally, for a given , an ellipse is the set

The midpoint of the line segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis, and the line perpendicular to it through the center is called the minor axis. It contains the vertices, which have distance to the center. The distance of the foci to the center is called the focal distance or linear eccentricity. The quotient is the eccentricity.

The case yields a circle and is included.

The equation can be viewed in a different way (see picture):
If is the circle with midpoint and radius , then the distance of a point to the circle equals the distance to the focus :

is called the circular directrix (related to focus ) of the ellipse.[1][2] This property should not be confused with the definition of an ellipse with help of a directrix (line) below.

Using Dandelin spheres one proves easily the important statement:

  • Any plane section of a cone with a plane, which does not contain the apex and whose slope is less than the slope of the lines on the cone, is an ellipse.

Ellipse in Cartesian coordinates[edit]


If Cartesian coordinates are introduced such that the origin is the center of the ellipse and the x-axis is the major axis and

the foci are the points ,
the vertices are .

For an arbitrary point the distance to the focus is and to the second focus . Hence the point is on the ellipse if the following condition is fulfilled

Remove the square roots by suitable squarings and use the relation to obtain the equation of the ellipse:

  • or solved for y

The shape parameters are called the semi-major and semi-minor axes. The points are the co-vertices.

It follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin.

Semi-latus rectum[edit]

The length of the chord through one of the foci, which is perpendicular to the major axis of the ellipse is called the latus rectum. One half of it is the semi-latus rectum. A calculation shows

The semi-latus rectum may also be viewed as the radius of curvature of the osculating circles at the vertices .


An arbitrary line intersects an ellipse at 0, 1 or 2 points. In the first case the line is called exterior line, in the second case tangent and secant in the third case. Through any point of an ellipse there is exactly one tangent.

The tangent at a point of the ellipse has the coordinate equation
A vector equation of the tangent is
  • with

Proof: Let be an ellipse point and the vector equation of a line (containing ). Inserting the line's equation into the ellipse equation and respecting yields:

In case of line and the ellipse have only point in common and is a tangent. The tangent direction is orthogonal to vector which is then a normal vector of the tangent and the tangent has the equation with a still unknown . Because

An ellipse (red) obtained as the intersection of a cone with an inclined plane
Ellipse: definition with director circle
shape parameters:
a: semi-major axis,
b: semi-minor axis
c: linear eccentricity,
p: semi-latus rectum.

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