Word Finder - ellipse

definition

A closed curve, the locus of a point such that the sum of the distances from that point to two other fixed points (called the foci of the ellipse) is constant; equivalently, the conic section that is the intersection of a cone with a plane that does not intersect the base of the cone.

definition

(grammar) To remove from a phrase a word which is grammatically needed, but which is clearly understood without having to be stated.

It is the envelope of circles described on the central radii of an ellipse as diameters.

The tricky part is making an ellipse at a point different from the origin.

When the ellipse becomes a circle, the meridian line becomes a straight line parallel to the axis, and the film passes into the form of a cylinder of revolution.

The Greek geometers were perfectly familiar with the property of an ellipse which in the Cartesian notation is x 2 /a 2 +y 2 /b 2 =1, the equation of the curve; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the curve.

The nature of the two kinds of branches is best understood by considering them as projections, in the same way as we in effect consider the hyperbola and the parabola as projections of the ellipse.

The angle cp is termed the eccentric angle, and is geometrically represented as the angle between the axis of x (the major axis of the ellipse) and the radius of a point on the auxiliary circle which has the same abscissa as the point on the ellipse.

When the ellipse differs infinitely little from a circle, the equation of the meridian line becomes approximately y = a+c sin (x/a) where c is small.

Arrow heads at the ends of an axis of an ellipse indicate tension as distinct from compression, and the semi-axes in magnitude and direction represent the principal stresses.

Above this entrance it widens into an ellipse a mile long, half a mile broad and 15 ft.

Newton replied promptly, "an ellipse," and on being questioned by Halley as to the reason for his answer he replied, " Why, I have calculated it."

Similarly any other property might be used as a definition; an ellipse is the locus of a point such that the sum of its distances from two fixed points (the foci) is constant, &c., &c.

Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or conics, the ellipse and the parabola consist each of one branch, the hyperbola of two branches.

As regards the so-called hyperbolisms, observe that (besides the single asymptote) we have in the case of those of the hyperbola two parallel asymptotes; in the case of those of the ellipse the two parallel asymptotes become imaginary, that is, they disappear; and in the case of those of the parabola they become coincident, that is, there is here an ordinary asymptote, and a special asymptote answering to a cusp at infinity.

Using a powerful and elaborate analysis, Adams ascertained that this cluster of meteors, which belongs to the solar system, traverses an elongated ellipse in 334 years, and is subject to definite perturbations from the larger planets, Jupiter, Saturn and Uranus.

This circle, projected in Q perspective as an ellipse, is shown in X the figure.

If projected with this velocity in any direction the point of projection will be at the end of the minor axis of the orbit, because this is the only point of an ellipse of which the distance from the focus is equal to the semi-major axis of the curve, and therefore the only point at which the distance of the body from the sun is equal to its mean distance.

He long adhered to the traditional belief that all celestial revolutions must be performed equably in circles; but a laborious computation of seven recorded oppositions of Mars at last persuaded him that the planet travelled in an ellipse, one focus of which was occupied by the sun.

Bessel announced, in December 1838, the perspective yearly shifting of 61 Cygni in an ellipse with a mean radius of about one-third of a second.

The orbit of the moon around the earth, though not a fixed curve of any class, is elliptical in form, and may be represented by an ellipse which is constantly changing its form and position, and has the earth in one of its foci.

The relation of the ellipse to the other conic sections is treated in the articles Conic Section and Geometry; in this article a summary of the properties of the curve will be given.

If the tangents be at right angles, then the locus of the point is a circle having the same centre as the ellipse; this is named the director circle.

The intersections of the lines drawn from corresponding points are points on the ellipse.

The thread is now stretched taut by a pencil, and the pencil moved; the curve traced out is the desired ellipse.

Let P be the intersection of the line SL with the line RAM, then it can be readily shown that P is a point on the ellipse.

Then with centre 0 1 and radius OJ, =OA 1, describe an arc. By reflecting the two arcs thus described over the centre the ellipse is approximately described.

In ancient geometry the name was restricted to the three particular forms now designated the ellipse, parabola and hyperbola, and this sense is still retained in general works.

One definition, which is of especial value in the geometrical treatment of the conic sections (ellipse, parabola and hyperbola) in piano, is that a conic is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio.

This ratio, known as the eccentricity, determines the nature of the curve; if it be greater than unity, the conic is a hyperbola; if equal to unity, a parabola; and if less than unity, an ellipse.

A conic may also be regarded as the polar reciprocal of a circle for a point; if the point be without the circle the conic is an ellipse, if on the circle a parabola, and if within the circle a hyperbola.

An important property of confocal systems is that only two confocals can be drawn through a specified point, one being an ellipse, the other a hyperbola, and they intersect orthogonally.

The definitions given above reflect the intimate association of these curves, but it frequently happens that a particular conic is defined by some special property (as the ellipse, which is the locus of a point such that the sum of its distances from two fixed points is constant); such definitions and other special properties are treated in the articles Ellipse, Hyperbola and Parabola.

When the cutting plane is inclined to the base of the cone at an angle less than that made by the sides of the cone, the latus rectum is greater than the intercept on the ordinate, and we obtain the ellipse; if the plane is inclined at an equal angle as the side, the latus rectum equals the intercept, and we obtain the parabola; if the inclination of the plane be greater than that of the side, we obtain the hyperbola.

Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the parabola, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle.

His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a conic, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a parabola.

He also considered the two branches of a hyperbola, calling the second branch the "opposite" hyperbola, and shows the relation which existed between many metrical properties of the ellipse and hyperbola.

We may also notice Kepler's approximate value for the circumference of an ellipse (if the semi-axes be a and b, the approximate circumference is ir(a+b)).

The major axis of the ellipse will be along the East-West line and the minor axis of the ellipse will be along the East-West line and the minor axis will be North-South.

The mean is trimmed since some incorrect ellipse fits produce extremely deviant parameter estimates that have a large influence on the mean.

Using the ellipse Tool, create a circular ellipse roughly 20 pixels in diameter on the left edge of the stage.

The shadow on the Earth's surface was a very elongated ellipse in these places.

The tendrils are moderately thin and long; one made a narrow ellipse in 5 hrs.

Principal strains, strain ellipsoid and strain ellipse; special types of strain ellipsoid and Flinn plots.

Thus it has a real centre, two foci, two directrices and two vertices; the transverse axis, joining the vertices, corresponds to the major axis of the ellipse, and the line through the centre and perpendicular to this axis is called the conjugate axis, and corresponds to the minor axis of the ellipse; about these axes the curve is symmetrical.

If the law of attraction is that of gravitation, the orbit is a conic section - ellipse, parabola or hyperbola - having the centre of attraction in one of its foci; and the motion takes place in accordance with Kepler's laws (see Astronomy).

But unless the orbit is an ellipse the body will never complete a revolution, but will recede indefinitely from the centre of motion.

Nevertheless, it should be observed that our globes take no account of the oblateness of our sphere; but as the difference in length between the circumference of the equator and the perimeter of a meridian ellipse only amounts to o 16%, it could be shown only on a globe of unusual size.

The major axis of any such aberrational ellipse is always parallel to AC, i.e.

As the ellipse degenerates into the straight line joining its foci, the contracted parts of the unduloid become narrower, till at last the figure becomes a series of spheres in contact.

About 3 metres in front of them was arranged a pair of smaller horizontal j aeroplanes, shaped like a long narrow ellipse, which formed the rudder that effected changes of elevation, the driver being able by means of a lever to incline them up or down according as he desired to ascend or descend.

The eccentricity of the ellipse is in the general average about 0.055, whence the moon is commonly more than i' further from the earth at apogee than at perigee.