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ellipse

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ellipse

1. 1. Ellipses are useful in drawing because they are perspective views of circles.However, the perspective projections may be difficult to draw, because majorand minor axes do not project as major and minor axes, centres do not projectto centres (in general), concentric circles do not project as concentric ellipses,and so on. Ellipses may be drawn from the definitions above by calculatingand plotting points, or directly by precise or approximate constructions. Theconcentric-circle method introduced above is one practial method forconstructing an accurate ellipse when the axes are known.A construction based on the focal property may be used for laying out smallellipses in the field, called gardeners ellipses. Place two stakes in the groundwith tacks in their tops marking the desired foci. Then make a loop of stoutcord of length 2a + 2c, and loop it around the tacks. Taking something like achaining pin or pointed metal rod, put it in the loop and draw the loop taut.Everywhere the pin may be will be a point on the ellipse, and as many pointsas desired may be marked. This method works quite well and is rapid. Stakesand string make it possible to lay out many decorative curves: circles, ellipses,parabolas and spirals, as the constructors of corn circles know.
2. 2. The simplest way to draw an ellipse on the drawing board is by using atrammel, as shown at the right. This is simply a straight strip of cardboard,with distances o-a = a and o-b = b marked on it. The perpendicular axes aredrawn where the ellipse is required, and the position of point "o" when theother points are carefully held on the axes is a point on the ellipse. There aremachines for drawing ellipses based on this principle, but they are expensiveand probably no longer manufactured. The trammel requires care, but is deadeasy to use.A standard method is called the parallelogram method, and is applicable toconjugate diameters as well as to the usual perpendicular axes. OE and EAare divided into N equal intervals. In the figure, N = 2 for simplicity. Then drawCB and intersect with DG extended to determine a point P on the ellipse. Asimilar method can be used to draw a parabola--it is all in how the intersectinglines are drawn. Note that we could draw circles with this method or thetrammel, but there are easier ways!
3. 3. The theory behind the parallelogram method can be found from the diagramat the right. The point P on the ellipse is the intersection of lines AB and CD.s is a parameter that runs from 0 to 1 as P goes from the end of the minoraxis to the end of the major axis. Line AB has the equation y = b - sbx/a,while line CD has the equation x = sa(b + y)/b. Eliminating s between thesetwo equations, we find quite readily that x2/a2+ y2/b2= 1, which is theequation of an ellipse with semimajor axis a and semiminor axis b.If you have the major minor axes of an ellipse, it is easy to find the foci byswinging an arc of radius equal to the semimajor axis from the end of aminor axis. This arc will cut the major axis at the focal points. Conversely,if you know the foci and the major axis, the intersection of arcs drawn fromthe foci with radius equal to the semimajor axis will determine the ends ofthe minor axis.
4. 4. The above methods all make accurate ellipses. It is much more convenient torepresent an ellipse by circular arcs that can be drawn with a compass than toconnect points laboriously determined. The simplest case is shown in thefigure at the left, called a three-center arch. To a draftsman, it is a "four-centerellipse." The centers are the symmetrically-placed D, C, D and C. On the lineAB joining the ends of the major and minor axes, lay off distance BM = a - b,the difference in the semiaxes. Now draw the perpendicular bisector of theremainder AM, the line L, which determines the centers D and C, as well asthe point G where the two arcs meet. Now the symmetrical points D and Ccan be laid out. Arcs of radii DG and CG are then drawn to complete the archor ellipse. This method is excellent for representing ellipses on drawings, butmay be a little crude for an actual arch, where the approximation may give anuneasy feeling. It would probably be better in any case to calculate an exactellipse and lay it out by coordinates for the actual structure, while the three-centered approximate ellipse will always do for a drawing.
5. 5. Stevens Method gives a somewhat better three-centred ellipse useful in isometricprojection. In the figure at the right, an ellipse is to be drawn in the rhombus ABCD,which is the isometric projection of a square. Using two triangles, first find the tangentpoint E, and draw an arc of radius R equal to ED, cutting the minor axis at Q. Do thesame thing with C as centre. With radius OQ, draw an arc locating points P and P onthe major axis. Now draw PD and extend it to the arc of radius R. This is not shown toavoid confusing the diagram. Now, draw arcs of radius r equal to the distance from Pto the curve at each end. They will be tangent to the arcs of radius R, so theapproximate ellipse has been constructed.A five-centred arch gives a better approximation by using three different radii,but is a little complicated to lay out. Here is the method: refer to the figure at theleft. Start with the box AFDO with width equal to the half-span and height equalto the rise. Draw AD, and then from F draw a perpendicular to this line, whichintersects the minor axis at H. Make OK equal to OD, and then draw a circle onAK as a diameter (Q is not the center of the line AK). Now lay off OM equal toLD, and draw an arc with center at H and radius HM. You will not have the pointN yet, but be patient. From A, lay off AQ equal to OL, and AP equal to half ofAQ. P should be on the line FH. Draw an arc with center P and radius PQ,intersecting the arc through M at point N. Now, P, N and H are the centers of thearcs defining one-quarter of the approximate ellipse. The sectors are shaded tomake this clear, and to show that the arcs are tangent to each other. Find thecenters of the remaining arcs by symmetry.
6. 6. For accurate drawing, it is useful to be able to find the directions of tangents toan ellipse, and points of tangency. Tangents to a circle are perpendicular to theradius at the point of tangency, so it is easy to draw a tangent at a given point ona circle, or to construct the tangent from an external point. The principles of twomethods for drawing tangents to an ellipse are shown in the figure at the right.The method at (a) uses the auxiliary circle. First, the point S on the auxilary circlecorresponding to the point of tangency P is found, and a tangent to the auxiliarycircle is constructed, cutting the axis at point T. The desired tangent is the lineTP. This method uses the vertical compression that turns the auxiliary circle intothe ellipse.Method (b) uses the focal properties of the ellipse. The focal radii from P to thefoci F and F are drawn, and the external angle is bisected. The bisector is thedesired tangent. This direction is perpendicular to the bisector of the internalangle between the focal radii.
7. 7. The fact that the tangent is perpendicular to the bisector of the angle betweenthe focal radii at P is not difficult to prove. In the figure at the right, theneighborhood of a point A on the ellipse is shown. Suppose B is a neighboringpoint, so close that the directions to F and F are not sensibly changed. For B tobe a neighboring point on the ellipse, the sum of the focal radii must remainunchanged (and equal to the major axis). In the direction shown, it is clear thatthe lengthening BC of the focal radius to F is equal to the shortening AD of thefocal radius to F, so that their sum remains constant, and B is also a point onthe ellipse, or, in this approximation, on the tangent from A. The tangent, ofcourse, is the limit of the chord through BA as B approaches A.The method for drawing a tangent from an external point P to an ellipse is shownat the right. First, draw a circle with centre at P and radius PF. The intersection ofthis circle with an arc of a radius equal to the major axis (2a) is point E. Point R isthe intersection of radius FE and the ellipse. In the triangles PER and PFR,corresponding sides are equal, since FR + RF = 2a = ER + RF, or FR = ER. PEand PF are equal by construction, while the side PR is common. Therefore, PRbisects the external angle and is thus the tangent. A similar construction givesthe other tangent from point P.
8. 8. The line PR also bisects the line FE. In the diagram, the triangles do not appearexactly congruent because the foci are not accurately located. If you make acareful drawing, the triangles will be congruent, and PR will be an accurateperpendicular bisector of FE. Since FE is perpendicular to the tangent, if we wanta tangent with a certain direction (and there is no point P given), a line is drawnin the perpendicular direction from F, and E is the intersection with an arc oflength 2a from F. Now EF is bisected, and the perpendicular bisector is thedesired tangent. We can now find tangents at a point on the ellipse, from anexternal point, and in a certain direction.If the focal points F and F approach one another, direction of the tangent is moreand more restricted. In the limit as the ellipse becomes a circle, F = F, and thetangent will be perpendicular to the radius, as we know from Euclid.The drawing utilities in Windows draw an ellipse in a circumscribed rectangle,which made preparing the graphics for this page quite easy, compared with theeffort required for parabolas and hyperbolas, which are not easy to draw with theWindows routines. It is annoying not to be able to draw a circle from center andradius, as in real drafting programs, but this can be worked around by putting thecenter of a square of side equal to the diameter at the desired center point. If theShift key is held down in a graphics program, it is only necessary to move thecursor along one diameter to get a circle, which makes it a little easier. Atemplate of three small crosses can be constructed and copied to wherever acircle is desired if you are drawing a number of circles of the same diameter. Onecross is the center, the other two the ends of a horizontal line equal to thediameter.
9. 9. The line PR also bisects the line FE. In the diagram, the triangles do not appearexactly congruent because the foci are not accurately located. If you make acareful drawing, the triangles will be congruent, and PR will be an accurateperpendicular bisector of FE. Since FE is perpendicular to the tangent, if we wanta tangent with a certain direction (and there is no point P given), a line is drawnin the perpendicular direction from F, and E is the intersection with an arc oflength 2a from F. Now EF is bisected, and the perpendicular bisector is thedesired tangent. We can now find tangents at a point on the ellipse, from anexternal point, and in a certain direction.If the focal points F and F approach one another, direction of the tangent is moreand more restricted. In the limit as the ellipse becomes a circle, F = F, and thetangent will be perpendicular to the radius, as we know from Euclid.The drawing utilities in Windows draw an ellipse in a circumscribed rectangle,which made preparing the graphics for this page quite easy, compared with theeffort required for parabolas and hyperbolas, which are not easy to draw with theWindows routines. It is annoying not to be able to draw a circle from center andradius, as in real drafting programs, but this can be worked around by putting thecenter of a square of side equal to the diameter at the desired center point. If theShift key is held down in a graphics program, it is only necessary to move thecursor along one diameter to get a circle, which makes it a little easier. Atemplate of three small crosses can be constructed and copied to wherever acircle is desired if you are drawing a number of circles of the same diameter. Onecross is the center, the other two the ends of a horizontal line equal to thediameter.
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