Fig. 11/1 shows the five sections that can be obtained from a cone The tnangle and the circle have been discussed in earlier chapters; this chapter looks at the remain -ing three sections, the ellipse, the parabola and the hyperbola These are three very important curves. The ellipse can vary m shape from almost a circle to almost a straight line and is often used in designs because of its pleasing shape The parabola can be seen m the shape of electnc fire reflectors, rader dishes and the main cable of suspension bridges Both the parabola and the hyperbola are much used in civil engineering. The immense strength of structures which are parabolic or hyperbolic m shape has led to their use in structures made of pre-cest concrete and where large unsupported ceilings are needed
ELLIPSE SECTION 3-3 (LESS ACUTE THAN SIOE OF CONE)
HYPERBOLA SECTION 5-5 (MORE ACUTE THAN SIOE OF CONE)
Fig. 11/1 Conic sections
SECTION 2-2 (HORIZONTAL)
HYPERBOLA SECTION 5-5 (MORE ACUTE THAN SIOE OF CONE)
Fig. 11/1 Conic sections
SECTION 2-2 (HORIZONTAL)
The height of the cone end the base diameter, together with the angie of the section relative to the side of the cone, are the factors which govern the relative shape ol any ellipses, parabolas or hyperbolas. There are an infinite number of these curves and. given eternity and the inclination to do so. you could construct them ell by taking sections from cones. There are other ways of constructing these curves and this chapter, as welt as showing how to obtain them by plotting them from cones, shows some other equally important methods of construction.
THE ELLIPSE Pig. 11/2 shows an ellipse with the important features labelled.
The ellipee m • conic section
Fig. 11/3 shows In detail how to project en ellipee as a section of a cone. The shape across X-X is an ellipse.
First draw the F.E.. the E.E. and the plan of the complete cone. Divide the pian into 12 equal sectors with a 60* set square. Project these sectors onto the F.E. and the E.E. where they appear as lines drawn on the surface of the cone from the base to the apex. The points where these linee croes X-X can be eesity projected across to the E.E.
and down to the pian to give the shape of X-X on these elevations. The point on the centre line must be projected onto the pian via the E.E. (follow the arrows).
The shape of X-X on the pian is not the true shape since, m the plan. X-X is sloping down into the pege. However, the widths of the points, measured from the centre line, are true lengths and can be transferred from the plan to the euxiliary view with dividers to give the true shape across X-X. This is the ellipse
TRUE ^CROSS
ANGLE OF X-X TO BASE IS LESS THAN THE SIDE OF THE CONE
TRUE ^CROSS
ANGLE OF X-X TO BASE IS LESS THAN THE SIDE OF THE CONE
The ellipse as a locus
An ellipse is the locus of a point which moves so that its distance from e fixed point (called the focus) bears a constant ratio, always less than 1, to its perpendicular distance from a straight line (called the directrix). An ellipee has two foci and two directrices.
Hg. 11/4 shows how to draw an aHips« given tha relative positions of the focus and tha directrix, and the eccentricity In this case the focus and the directrix are 20 mm apart and the eccentricity is J.
The first point to plot is the one thet lies between the focus and the directrix. This is done by dividing OF in the same ratio as the eccentricity. 4: 3. The other end of the ellipee. point P. is found by working out the simple algebraic sum shown on Fig. 11/4.
The condition for the locus is that it ia always 4 as far from the focus es it is from the directrix. It is therefore j as fsr from the directrix as it is from the focus Thus, if the point is 30 mm from F, it is If mm from the directrix; if the point is 20mm from F it is fx20mm from the directrix; if the point is 30 mm from F. It is J x 30 mm from the directrix. This is continued for as many points as may be necessary to draw an accurate curve. The intersections of radii drawn from F and lines drawn parallel to the directrix, their distance from the directrix being proportional to the radii, give the outline of the ellipse These points ere joined together with a neat freehand curve.
STAGE 3
Fig 11/5
To construct an ellipse by concentric circles We now come to the first of three simple methods of constructing an ellipse. All three methods need only two pieces of information for the construction—the lengths of the major and minor axes.
Stage 1. Draw two concentnc circles, radii equal to \
major and } minor axes. Stege 2. Divide the cirde into a number of sectors. If the ellipee is not too large, twelve will suffice as shown in Fig. 11/5.
Fig 11/5
Stage 3. Where the sector lines cross the smaller circle, draw horizontal lines towerds the larger circle. Where the sector lines cross the larger circle, draw vertical lines to meet the horizontal lines. Stage 4. Draw a neat curve through the intersections.
Stage 1. Draw a rectangle, length and breadth equal to the mejor and minor exe» Stage 2. Divide the two »horter »idea of the rectangle into the »ame tvtn number of equal part*. Divide the major axis into the »ame number of equal parts Stage 3. From the point» where the minor axis crosses the edge of the rectangle, draw intersecting lines as shown in Fig. 11 /6. Stage 4. Draw a neat curve through the intersections.
To construct en ellipse with a trammel A trammel is a piece of stiff paper or card with a straight edge. In this case, mark the trammel with a pencil ao that half the major and minor axes are marked from the same point P. Keep B on the minor axis. A on the major axis and slide the trammel, marking at frequent intervals the position of P. Fig. 11/7 shows the trammel in position for plotting the top half of the ellipse; to plot the bottom
NORMAL
TANGENT
Fig 11/8
To find the foci, the normal and tha tangent of an ellipse (Fig 11/8)
The foci. With compasses sat at a radius of J major axis, centre at the point where the minor exit crosses the top
(or tha bottom) of tha ellipse, strike an arc to cut the major axis twice. These are the foci.
NORMAL
TANGENT
Fig 11/8
3RD ANGLE PROJECTION
11/9
Y-Y IS PARALLEL TO THE SIDE OF THE CONE
The norma! at my point P. Draw two lines from P. one to each focus and bisect the angle thus formed. This bisector is a normal to the ellipse
Tha tangant at any point P. Since the tangent and normal are perpendicular to each other by definition, construct the normal and erect a perpendicular to it from P. This perpendicular is the tangent.
THE PARABOLA The parabola aa a conic section The method used for finding the ellipse in Fig 11/3 can be adapted for finding a parabolic section. However, the method shown below is much better because it allows for many more points to be plotted. In Fig. 11/9 the shape across Y-Y is a parabola.
First divide Y-Y into a number of equal pans, in this case 6. The radius of the cone at each of the seven spaced points is projected on to the plan and circles are drawn. Each of the points must lie on its respective circle The exact position of each point is found by projecting it onto the plan until it meeta its circle. The points can then be joined together on the plan with a neat curve.
The E E is completed by plotting the intersection of the projectors of each point from the F.E. and the plan.
Neither the E.E. nor the plan show the true shape of Y-Y since, m both views. Y-Y Is sloping into the paper The only way to find the true shape of Y-Y is to project a view at right angles to it. The width of each point measured from the centre line, can be transferred from the plan as shown.
3RD ANGLE PROJECTION
11/9
Y-Y IS PARALLEL TO THE SIDE OF THE CONE
The para bola as a locus
A parabola it the locus of a point which move» to thai its distance from a fixed point (called the focut) beat a constant ratio oM to its pe<pendicular distanca from a ttrrght I ma (called the directnx).
Fig 11/10 snow« how to draw a pa'tbola givan the relative potitiont ol the focut and the directnx. In th«» case th« focus and directrix ere 20 mm apart.
The first point to plot it the one that baa between the toe us and the directnx. By definition it It the tame dtstanca, 10 mm. from both.
The condition for the loco» - «het it « always the same distance from the focut at it it from the directrix The parabola is therefore found by plotting the intersections of radii 15 mm, 20 mm. 30 mm. etc. centre on the focut. with lines drawn parallel to the directnx at distances IB mm. 20 mm. 30 mm, etc.
3RD ANGLE PROJECTION
Fig 11/12
Fig. 11/13
ANGLE OF Z-Z TO THE BASE IS GREATER THAN THE SIDE OF THE CONE
ANGLE OF Z-Z TO THE BASE IS GREATER THAN THE SIDE OF THE CONE
To find tha focus of a parabola and tha tangent at a point P
Salad a point R on tha axis which is obviously further from V than the locus will be. From R erect a perpendicular and mark off RS — 2VR. Jotn SV; this cuts the parabola in T. From T drop a perpendicular to meet the axis in F. F is the focus.
To draw the tangent at P. join FP and draw PQ parallel to the axis. The bisector ol FPQ is the tangent
THE HYPERBOLA The hyperbola aa a conic section The method is identical to that used for finding the pereboUc section in Fig. 11/9. The construction. Fig. 11/13. can be followed from the instructions for that figure.
Fig 11/12
Fig. 11/13
The hyperbola as a locus
A hyperbola is the locus of a point which moves so that its distance from a fixed point (called the focus) bears a constant ratio, always greater than 1. to its perpendicular distance from a straight line (called thedirectnx).
Fig. 11/14 shows how to draw a hyperbola given the relative positions of the focus and the directrix (in this case 20 mm) and the eccentricity (3/2).
The first point to plot is the one that lies between the focus and the directnx. This is don« by dividing the distance between them in the same ratio as the eccentricity. 3:2.
The condition for the locus is that it is always { as far from the directrix as it is from the focus Thus, if the point is 15mm from the focus*, it is fx 15mm from the directnx; if it is 20 mm from the focus, it is j x20 mm from the directrix. This is continued for as many points as may be required
There are constructions for the normal and tangent to a hyperbola but they introduce additional features which ara beyond the scope of this book.
Exercises 11 (All questions originally set in Imperial units) 1 Fig. 1 is the frustum of a right cone. Draw this elevation and a plan Draw th« true shap« of th« feca AB Southern Regional Examinations Board
2. Fig. 2 shows a point P which moves so that the sum of the distance from P to two fixed points. 100 mm apart is constant and equal to 125 mm. Plot the path of the point P. Name the curve end the given fixed points.
Associated Lancashire Schools Examining Board np.px-izs—
Fig 11/14
ECCENTRICITY -f DIMENSIONS IN mm
Fig 11/14
3. Fig. 3 shows the loud-speaker grill of a car radio. The grill is rectangular with an elliptical hole. Draw the grill, full size, showing the construction of the ellipse clearly
West Midlands Examinations Board
14o-
4. Fig 4 shows an elliptical fish-pond for a small garden The ellipse is 1440 mm long and 720 rem wide. Using a scale of draw a true elliptical shape of the pond. (Do not draw the surrounding stones.) AH construction must be shown. If a paper trammel is used, en accurate drawing of it must be made.
5. Fig. 6 shows a section, based on en ellipse, for a handrail which requires cutting to form a bend so that the horizontal overall distance is increased from 112 mm to 125 mm.
Construct the given figures and show the tangent construction at P and P,.
Show the true shape of the cut when the horizontal distance is increased from 112 mm to 125 mm. Southern Universities' Joint Board (See Ch. 7 for information not in Ch. 11 )
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DIMENSIONS IN mm 6. Fig. 6 shows the upper hall of the section of a small headlamp. The casing is in the form of a semi-ellipse F is the focal point. The reflector section is in the form of a parabola. Part 1. Draw, full site, the complete semi-ellipse. Part 2. Draw, lu« site, the complete parabola inside the semi-ellipse Southern Regional Examinations Board7. A point moves in a plane in such a way that its distance from a fixed point is equal to its shortest distance from a fixed straight line. Plot the locus of the moving point when the fixed point is 44 mm from the fixed line. The maximum distance of the moving point is 125 mm from the fixed point. State the name of the locus, the fixed point and the fixed line. Associated Examining Board 8 A piece of wire is bent into the form of a parabola. It fits into a rectangle which has a base length of 125 mm and a height of 100 mm. The open ends of the wire are 125 mm apart. By means of a single line, show the true shape of the wire. Cambridge Local Examinations 9. An arch has a span of 40 m and a central rise of 13 m and the centre line is an arc of a parabola. Draw the centre line of the arch to a scale of 10 mm -20 m. Oxford and Cambridge Schools Examination Board 10. A cone, vertical height 100 mm. base 75 mm diameter, is cut by a plane parallel to its axis and 12 mm from it Draw the necessary views to show the true shape of the section and state the name of it. Oxford and Cambridge Schools Examination Board 11. Fig. 7 shows a right cone cut by e plane X-X. Draw the given view end project an elevation seen from the left of the given view. 12. Draw the conic having an eccentricity of | and a focus which is 38 mm from the directnx. State the name of this curve. Associated Examning Board DIMENSIONS M mm DIMENSIONS M mm |
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