Ellipse
An ellipse is a geometric shape that resembles a flattened circle, characterized by two foci and a major and minor axis. In mathematics, it is defined as the set of all points where the sum of the distances to the two foci is constant. The equation of an ellipse in the coordinate plane is commonly represented in standard form.
6 Key excerpts on "Ellipse"
eBook - ePub- Michael Harrison, Patrick Waldron(Authors)
- 2011(Publication Date)
- Routledge(Publisher)
...the Ellipse. The equation of the Ellipse takes its simplest form when the focus S is a point on the negative x axis, say(− aϵ, 0), where a > 0, and the directrix is the vertical line with equation x =− a/ϵ. Such an Ellipse is depicted in Figure 4.2. The square of the distance of P from the focus S is (x + aϵ) 2 + y 2 and the square of the distance of P from the directrix L is (x + a/ϵ) 2. Thus the equation of this Ellipse is (x + a ϵ) 2 + y 2 = ϵ 2 (x + a ϵ) 2 (4.5) Gathering up terms in x 2 and terms in y 2 and noting that identical terms in x on each side cancel, this simplifies to x 2 (1 − ϵ 2) + y 2 = a 2 (1 − ϵ 2) (4.6) or x 2 a 2 + y 2 a 2 (1 − ϵ 2) = 1 (4.7) If we. define b by b 2 = a 2 (1 − ϵ 2) (which we can do because we have assumed that ϵ < 1, which guarantees that b is not imaginary), then the equation of the Ellipse becomes x 2 a 2 + y 2 b 2 = 1 (4.8) Note that by construction b < a. Note also that this equation contains only even powers of both x and y, so that the Ellipse must be symmetric about both coordinate axes. From this symmetry, we can deduce the existence of a second focus S ′ at (a ɛ, 0) and a second directrix, the line L ′ with equation x = a /ɛ. When x = 0, y = ± b, and when y = 0, x = ± a, so the Ellipse cuts the coordinate axes in the points (a, 0), (0, b), (− a, 0) and(0, − b), as indicated. The longer (horizontal) axis of the Ellipse, which is of length 2 a, is called the major axis ; the shorter (vertical) axis, which is of length 2 b, is called the minor axis. As the eccentricity approaches unity, b approaches zero, so the Ellipse collapses onto the x axis. Figure 4.2 Ellipse with foci (±aϵ, 0) and directrices x = ±aϵ Finally, note that, for any value of the angle ϕ, the point (a cos ϕ, b sin ϕ) lies on the parabola, since sin 2 ϕ + cos 2 ϕ = 1...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...13.5 b. Taking as case study the first possibility – and remembering that F 1 (−c, 0), P eq, 1 (0,b), and C (0,0), one attains (13.75) given the quantitative definition of distance between two given points via their Cartesian coordinates; after revisiting Eq. (13.52) as (13.76) Eq. (13.75) may be redone to (13.77) to eventually reach (13.78) Since b < a due to the definition of a and b as major and minor axes, respectively, 0 < ε < 1 in the case of an Ellipse – while ε = 0 is found for a circle, since a = b = R in this case (as plotted above); the eccentricity may accordingly be viewed as a measure of how far the Ellipse deviates from being circular. Conversely, a parabola is characterized by ε = 1, and a hyperbola by ε > 1. The Ellipse was first studied by Menaechmus (who died 320 BCE), investigated much later by Euclid, but named by Apollonius (who died c. 190 BCE); the concept of foci was introduced by Pappus of Alexandria (290–350 BCE). All conics abide to (13.79) as general Cartesian form – with coefficients being real numbers, and not all A, B, and C being equal to zero (otherwise a straight line would result); which type of conic is at stake depends on the sign of discriminant Δ ≡ B 2 − 4 AC. In fact, Δ < 0 stands for an Ellipse, since Eq. (13.53) becomes (13.80) after multiplying both sides by a 2 b 2 and moving a 2 b 2 to the left‐hand side; this leaves A = b 2, B = D = E = 0, C = a 2, and F = −a 2 b 2, as well as Δ = 0 2 − 4 b 2 a 2 = − 4 a 2 b 2 < 0. If A = C = 1, B = D = E = 0, and F = −R 2, then a circle of radius R results, see Eq. (13.79) vis‐à‐vis with Eq. (13.37), again with Δ = 0 2 – 4 = −4 < 0; a parabola is associated with Δ = 0, and a hyperbola with Δ > 0. 13.4 Length of Line Consider a generic curve, y ≡ y { x }, laid on the x 0 y plane, as depicted in Fig. 13.6...
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...Thus, the parabola in Figure 4.7 has the equation x - h = − (y - k) 2 4 p. FIGURE 4.7. Parabola with vertex at (h, k) and axis parallel to the x-axis. 7. Ellipse An Ellipse is the set of all points in the plane such that the sum of their distances from two fixed points, called foci, is a given constant 2 a. The distance between the foci is denoted 2 c ; the length of the major axis is 2 a, whereas the length of the minor axis is 2 b (Figure 4.8) and a = b 2 + c 2. FIGURE 4.8. Ellipse; since point P is equidistant from foci F 1 and F 2 the segments F 1 P and F 2 P = a ; hence a = b 2 + c 2.. The eccentricity of an Ellipse, e, is < 1. An Ellipse with center at point (h, k) and major axis parallel to the x-axis (Figure 4.9) is given by the equation (x − h) 2 a 2 + (y − k) 2 b 2 = 1. FIGURE 4.9. Ellipse with major axis parallel to the x-axis. F 1 and F 2 are the foci, each a distance c from center (h, k). An Ellipse with center at (h, k) and major axis parallel to the y-axis is given by the equation (Figure 4.10) (y − k) 2 a 2 + (x − h) 2 b 2 = 1. FIGURE 4.10. Ellipse with major axis parallel to the y-axis. Each focus is a distance c from center (h, k). 8. Hyperbola (e > 1) A hyperbola is the set of all points in the plane such that the difference of its distances from two fixed points (foci) is a given positive constant denoted 2 a. The distance between the two foci is 2 c and that between the two vertices is 2 a. The quantity b is defined by the equation b = c 2 − a 2 and is illustrated in Figure 4.11, which shows the construction of a hyperbola given by the equation x 2 a 2 − y 2 b 2 = 1. When the focal axis is parallel to the y-axis the equation of the hyperbola with center (h, k) (Figures 4.12 and 4.13) is (y − k) 2 a 2 − (x − h) 2 b 2 = 1 FIGURE 4.11. Hyperbola; V 1, V 2 = vertices; F 1 F 2 = foci. A circle at center O with radius c contains the vertices and illustrates the relation among a, b, and c...
eBook - ePubManual of Engineering Drawing
Technical Product Specification and Documentation to British and International Standards
- Colin H. Simmons, Dennis E. Maguire(Authors)
- 2012(Publication Date)
- Butterworth-Heinemann(Publisher)
...Chapter 12 Loci Applications If a point, line, or surface moves according to a mathematically defined condition, then a curve known as a locus is formed. The following examples of curves and their constructions are widely used and applied in all types of engineering. Methods of Drawing an Ellipse Two-circle Method Construct two concentric circles equal in diameter to the major and minor axes of the required Ellipse. Let these diameters be AB and CD in Fig. 12.1. FIGURE 12.1 Two-circle construction for an Ellipse. Divide the circles into any number of parts; the parts do not necessarily have to be equal. The radial lines now cross the inner and outer circles. Where the radial lines cross the outer circle, draw short lines parallel to the minor axis CD. Where the radial lines cross the inner circle, draw lines parallel to AB to intersect with those drawn from the outer circle. The points of intersection lie on the Ellipse. Draw a smooth connecting curve. Trammel Method Draw major and minor axes at right angles, as shown in Fig. 12.2. FIGURE 12.2 Trammel method for Ellipse construction. Take a strip of paper for a trammel and mark on it half the major and minor axes, both measured from the same end. Let the points on the trammel be E, F, and G. Position the trammel on the drawing so that point F always lies on the major axis AB and point G always lies on the minor axis CD. Mark the point E with each position of the trammel, and connect these points to give the required Ellipse. Note that this method relies on the difference between half the lengths of the major and minor axes, and where these axes are nearly the same in length, it is difficult to position the trammel with a high degree of accuracy...
eBook - ePubManual of Engineering Drawing
British and International Standards
- Colin H. Simmons, Dennis E. Maguire, Neil Phelps(Authors)
- 2020(Publication Date)
- Butterworth-Heinemann(Publisher)
...The distances from the focus are all radial, and the distances from the directrix are perpendicular, as shown by the illustration. Fig. 14.6 (A) Ellipse construction. (B) Hyperbola construction. To assist in the construction of the Ellipse in Fig. 14.5, the following method may be used to ensure that the two dimensions from the focus and directrix are in the same ratio. Draw triangle PA1 so that side A1 and side P1 are in the ratio of 3–5 units. Extend both sides as shown. From any points B, C, D, etc., draw vertical lines to meet the horizontal at 2, 3, 4, etc.; by similar triangles, vertical lines and their corresponding horizontal lines will be in the same ratio. A similar construction for the hyperbola is shown in Fig. 14.6. Commence the construction for the Ellipse by drawing a line parallel to the directrix at a perpendicular distance of P3 (Fig. 14.6 (a)). Draw radius C3 from point F1 to intersect this line. The point of intersection lies on the Ellipse. Similarly, for the hyperbola (Fig. 14.6 (b)) draw a line parallel to the directrix at a perpendicular distance of Q2. Draw radius S2, and the hyperbola passes through the point of intersection. No scale is required for the parabola, as the perpendicular distances and the radii are the same magnitude. Repeat the procedure in each case to obtain the required curves. Interpenetration Many objects are formed by a collection of geometrical shapes such as cubes, cones, spheres, cylinders, prisms, and pyramids., and where any two of these shapes meet, some sort of curve of intersection or interpenetration results. It is necessary to be able to draw these curves to complete drawings in orthographic projection or to draw patterns and developments. The following drawings show some of the most commonly found examples of interpenetration...
eBook - ePubThe Complete Guide to Perspective Drawing
From One-Point to Six-Point
- Craig Attebery(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...But there are two vanishing points—which one does the minor axis connect to? It is helpful to think of the minor axis as an axle on a tire. The minor axis—like an axle—goes through the Ellipse. The minor axis is a three-dimensional form. It is 90° from the surface of the Ellipse (Figure 8.15). Figure 8.15 When drawing vertical two-point Ellipses, the minor axis connects to a vanishing point. Think of the Ellipse as a tire, and the minor axis as an axle. Figure 8.16 To draw tapered cylindrical forms, measure the diameter on the ground plane, then project the Ellipse to the desired height and connect the Ellipses. Tapered Forms: Cups, Bottles, and the Like Figure 8.17 Create squares on the ground plane, raise the squares to the desired height, draw Ellipses, and then follow the contour to create a curved cylindrical shape. To draw cylindrical forms with various diameters, first draw squares of the appropriate size on the ground plane. Raise each square to the desired height. Then draw an Ellipse in each square. For example, a simple tapered cup will have a smaller diameter Ellipse on the ground and a larger diameter Ellipse above. Draw both on the ground (Figure 8.16, top). Then project the top of the cup to the desired height. Connect the two Ellipses to create the cup (Figure 8.16, bottom). For complex forms of varying diameters, make more Ellipses. The Ellipses serve as key cross-sections and guide the contour of the form— the more Ellipses, the more accurate the shape (Figure 8.17). Spheres Drawing a sphere in perspective is more complicated than one might think. A compass can be used to draw a circle, and considered finished. But if a specific size or placement for the sphere is desired, then a cube must be drawn first. The sphere fits into the cube touching the center of all six sides. The cube defines where the sphere touches the ground...
