Video:
Hello, everyone. Welcome to our NPTEL online certification courses on Engineering Drawing
and Computer Graphics. We are in module number 2, lecture number 14 on Conic Sections.
(Refer Slide Time: 00:25)
(Refer Slide Time: 00:28)
In this lecture, we have covered the focus-directrix method, concentric circle method, rectangle
method or oblong method and in today's class, we will look at the arc of the circle method.
(Refer Slide Time: 00:43)
In this arc of the circle method, an ellipse with the major axis is known perhaps let us say in this
case 100 mm, and the distance between foci is now something like 70 mm. If that is the case,
how to construct an ellipse? So, let us look at this steps, first of all, we have to draw major axis
AB 100 mm and locate the midpoint O. Draw major axis AB 100 mm and locate midpoint O.
So, point A, point B this is 100 mm, and at 50 mm we are going to locate O.
(Refer Slide Time: 01:50)
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Once it is done, locate foci, locate foci F 1 and F 2. Those are F 1 and F 2 on AB such that F 1 O
and O F 2 are 35 mm each. The major axis is 100 mm, and the distance between foci is 70 mm;
that means, we have to mark a division from O, here this is 35 units and this one also 35 units.
Our standard practice is we do not show these dimensions on the drawing somewhere outside we
have to extend those lines and show it. So, once we locate this F 1 foci, F 2 focus, then we will
go with the next step.
(Refer Slide Time: 02:54)
Now, we have to mark the suitable number of points something like 1, 2, 3, on AB between F 1
and F 2. F 1 and F 2 it can be equal or any number of points. It is easy always to go with an equal
number of points as a uniform grid first point, second point, third point, fourth point.
Once it is done with F 1 as centre; so, F 1 is centre the distance from A to 1, this distance let us
measure it with that distance from F 1 make an arc something like that on both the sides. So, the
centre has to be F 1, but the distance is A 1 A to one whatever the distance make an arc on top
and bottom on either side of AB.
Now, with focus F 2 from here, as a centre and radius B 1; so, B is here B 1 is that. So, with B to
1 distance, but the centre is F 2 make an arc in this direction; similarly, make it on that side.
(Refer Slide Time: 04:44)
Once we are done we will have this point P 1 and P 1’, these are F 1, and this is F 2, and this is
also F 2, and this is F 1.
Now, if we are moving to the second point to construct P 2 what we have to do is from A to 2
whatever the distance use that distance, but centre as F 1 makes an arc on either side. Similarly,
from B to 2 distance whatever the distance F 2 as centre make an arc on both sides. Whatever the
intersection point let us call P 2, P 2’ or P 2’ in such a way that A to 1 distance plus B to 1
distance always gives us major axis 70 mm.
Let us look at this. This is A 1; A 1 and B 1 is this. So, if we are going to add these two distances,
it is supposed to give us major axis 70 mm.
(Refer Slide Time: 06:13)
Similarly, if we are using A to 2 distance and B to 2 distance that also consists of A 2 plus B 2
distance that is also 70 mm; in that way, we will be going to construct P 2 point P 3 point and so
on things.
So, let us begin our example construct it step by step. Major axis 100 mm. So, let us look at on
this drawing sheet. So, let us mark 100 mm on this drawing sheet.
(Refer Slide Time: 07:01)
Let us call A and B; then we have to locate the centre which bisectors we can use it otherwise 50
mm also we can use in this case. Join these lines. So, this is the centre of that circle call O. Now,
we have to locate foci which is at 35 mm on either side. So, let us mark 35 mm scale. So, this is
one of the foci; the other one is at 70 mm. So, call these points F 1 and this point F 2 after that
we have to take distance A to 1; for that, we have to arbitrarily or equal divisions we can take it
to let us take equal divisions. So, it is 35 mm. So, we would like to mark after every 7 points. So,
35, 7 5s are 35 so, 6, 7, 14, 21, 28. Similarly, here one can make 35 next 42, 49 and so on. Once
this division is done, we have to measure distance A to 1, let us name them as 1, 2, 3, 4, 5, 6 and
so on. A to 1 whatever the distance use your compass A 1 use F 1 as centre mark an arc on both
sides from B to 1 also we have to find the distance. Use foci as F 2 make an arc, call that
intersection point as P 1 P 1, P 1’, P 1’. Now, A to 2; pick that point; this is A to 2 centred around
the focus on both sides then B to 2 whatever the distance we get use F 2 intersect this curve which
is A 2. So, mark this point P 2. Similarly, use A 3 distance centre around F 1 B 3 distance centred
around F 2 Now, we will be in a position to join a smooth curve using French curve it will be
better and so on. Similarly, we will be in a position to construct a line which passes through P 3.
In that way, we will be in a position to construct all that curve which pass through B point and
also at this point, so that we will be in a position to construct ellipse. This is by arcs of the circle
method.
In this case, we require a major axis. So, if it is construction lines we have to show 100 mm; foci
distance also we know. So, smaller dimensions are always inside. There is this standard
convention what we are going to use. The construction procedure can be much lighter also instead
of darkening it. So, this one is 70, and this one is 100, the distance between foci.
(Refer Slide Time: 14:39)
Now, we will ask a question: how to draw a tangent and normal to an ellipse. For example, part
of the curve we have constructed as an ellipse. Now, pick any point where we would like to
construct a normal or tangent. In this case, we would like to construct, for example, at point R.
So, this is the ellipse which we have joined, and we intend to construct something like normal to
that. For that purpose, what we have to do?
(Refer Slide Time: 15:24)
After deciding where you would like to draw the normal or tangent connect that point R with
focus F 1. Similarly, join this R with focus F 2. Now, we have an angle. We have this angle F1-
R-F2. Now, based on this centre somewhere distance we just make an arc we have to bisect it.
So, from there again with the same radius make an arc on both sides. So, we will have some point
and extend that. So, it makes equal angles on both sides, and this point extended. If we can do
that, that is what we call normal. So, to an ellipse, this is the way we constructed normal; tangent
always be 90° and touch at only one point passing through R, and we will be in a position to
construct tangent.
(Refer Slide Time: 16:52)
If some other point here then we will join F 1 point, for example, let us look at we would like to
construct a normal there. What we do is connect this F 1 and this point similarly connect this F
2. Now, make a bisector join it, construct normal, this is normal, and that will be tangent.
If one would like to construct normal here, what to be done, think about it. Precisely on the major
axis or minor axis, you would like to construct normal, will that be straightforwardly this one or
do I have to join the lines from F 1 F 2 and bisect it?
(Refer Slide Time: 17:50)
So, let us look at it carefully. If the point is straight away at the major or major axis, the procedure
is to connect any point on that curve with two foci. So, if I am going to connect point B with F
1, the line will be this one. Similarly, this point we are going to connect with F 2; that means, for
this point B again connect that. Both are still making that 0° or 180° kind of thing the equal
division which comprises of the same line.
So, this is the one which makes normal to that circle at this point, and tangent will be
perpendicular to that this will be. The same thing happens here normal will be in that direction
tangent will be in that direction. If we are picking C point normal will be in that direction, tangent
will be in another direction. Except for these four points A, B, C, D at remaining points, these
normal angles continuously increases as you go in that direction from 0 angles to 90° it increases.
So, let us construct this normal on a tangent on sheet. So, we have a part of the curve here on the
drawing sheet. Now, let us identify a point here.
Let us consider this is the point where I would like to construct normal on the drawing sheet here.
Now, the procedure is to join this F 1 with that point it is a construction line; similarly, construct
join this F 2 with that particular point. Now, this is the point what we are identifying.
Now, use angle bisector division for that purpose what we have to do is pick anything like
arbitrary radius. Now, use this intersection point with the same radius, intersect this curve locate
this point and join this point with this one. This will be the normal M N and the tangent always
be perpendicular to that.
So, we can use a set of squares these set squares we can use it, align that or perhaps your drafter
is the best way to construct these perpendicular lines and extend this line. So, we have this tangent
S T, the arc of the circle method, and we have learnt how to construct both tangent and normal.
(Refer Slide Time: 21:43)
So, in today's lecture, we have covered this arc of circles method. So, in principle to construct
ellipse, the popular methods are four - focus-directrix method, a concentric circle method, oblong
method and arc of circle method. In focus-directrix method we know something like a directrix,
eccentricity we know, equally divide the distance between these any point on the curve by an
equal number of divisions, draw 45° line and intersect it and construct this ellipse.
In the concentric circle method, we have a major axis, minor axis and two circles we draw, two
concentric circles by horizontal and vertical projections on an equal number of divisions we will
construct these ellipse. In rectangle method or oblong method, what we have is major axis, minor
axis, a rectangle; in some cases parallelogram also we will use to construct an equal number of
divisions on the major axis, minor axis. Connect minor axis elements with major axis elements
and get the intersection lines from there construct this ellipse.
In the arc of the circle method, we have seen major axis is given, the distance between foci is
given. So, using a principle any point where we would like to construct these ellipse A 1 plus B
1 always be major axis length using that principle and radius is F 1 F 2 the foci centres, make
many arcs intersect them and construct an ellipse this is. These are the ways we construct ellipse.
And especially focus-directrix method is quite popular: one because from directrix we know the
distance and eccentricity we know; that means, in the last classes we have seen an ellipse has
eccentricity less than 1, is for parabola eccentricity is equal to 1 and for hyperbola, eccentricity
is greater than 1. That means if we know the directrix distance and eccentricity, in principle, we
will be in a position to construct it can be ellipse, parabola, hyperbola.
(Refer Slide Time: 24:17)
And, in next class, we will look at these four methods especially focus-directrix method,
rectangle method and there is a particular case like the tangent method to construct parabolas.
We will also see how to draw tangent and normal to parabolas.
Thank you
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