Summary:
1. What is Hyperbola and how it’s formed?
2. Use of Hyperbola.
3. Methods to draw a Hyperbola.
4. What are Asymptotes?
5. How to draw or find out asymptotes in Hyperbola?
6. How to draw a Tangent and Normal on a Hyperbola?
What is Hyperbola or Hyperbolic Curve and how it’s generated?
When a curve is generated by cutting a conic section plane parallel to its axis, it is called a Hyperbolic Curve.
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As we know, a Hyperbola is a plane curve only which has two separate branches. It means a Hyperbolic curve is generated in two ways.
(i) Either, when two cones are pointing toward each other and both cones are intersected by a plane that is parallel to the axes of those cones,
(ii) Or, when the cutting plane of a conic section is parallel to the axis of the conic.
For simplicity, the cutting plane must be parallel to the axis to generate a Hyperbolic curve.
Use of Hyperbola
Practically, in engineering drawing, the Hyperbolic curves are used to draw Quenching towers, Hydro power plants, Water treatment plants, Water channels, etc.
Methods of Construction of Hyperbola
(1) Eccentricity Method or General Method or Common Method:
It’s similar to drawing an ellipse or a parabola in the Eccentricity
method. Taking the assumption, draw a Hyperbola, where the distance between the
directrix to focus is 50 mm and the eccentricity is equal to 3/2.
(i) Draw the directrix AB and the axis CD.
(ii) Mark the focus
F on CD and 50 mm from C.
(iii) Divide CF into 5 equal divisions, and mark V at the vertex point on the second division point from C. (As the eccentricity = 3/2, therefore, VF = 3 unit divisions, and CV
= 2 unit divisions.)
(iv) At vertex ‘V’, draw a perpendicular VE equal to VF.
(v) Now, join CE by a straight line and extend it to a
convenient length.
(vi) Mark points 1,
2, 3, and 4 on the axis and through it, draw perpendiculars to meet CE and
intersect at 1', 2',
3', and 4' respectively.
(vii) Now, take the center as F and radius equal to
1-1', and draw arcs intersecting the 1-1' perpendicular at P1 and P'1 on both sides of the axis.
(viii) Similarly, take
the center as F, and take the radius equal to 2-2', 3-3', and 4-4', and draw arcs intersecting
one by one to the 2-2', 3-3', and 4-4' perpendiculars to get the points P2 and P'2, P3 and P'3, P4 and P'4 respectively.
(ix) Now, draw a
smooth thick curve through these points V, P1, P'1, P2, P'2, P3, P'3, P4, and P'4 to get the
required hyperbola by Eccentricity or General or Common method.
(2) Hyperbola drawn by Rectangle Method
When a Hyperbola
needs to be drawn through a given point positioned between two lines making a right
angle (90˚) between them, it can be easily drawn with this method.
(i) First of all, draw two straight lines OA and OB, at right angles (90˚) to each other.
(ii) Take the point P
anywhere in the quadrant or as per given. See image.
(iii) Now, through point P, again draw two lines in such a way, that CD is parallel to OA and EF parallel to OB.
(iv) Mark some points 1, 2, 3, 4, and 5 on line PD, and some points 6, 7, and 8 on line PC, as per our assumption or requirements, equidistance is not mandatory.
(v) Now, join O1, O2, O3, O4, and O5 by straight lines which intersect the PF line at points 1', 2', 3', 4', and 5'.
(vi) Similarly, join O6, O7, and O8 by straight lines, and extend those to intersect the line PE at points 6', 7', and 8'.
(vii) Now, draw vertical parallel lines to OB, through the points 1, 2, 3, 4, 5, 6, 7, and 8. And draw horizontal parallel lines to OA, through 1', 2', 3', 4', 5', 6', 7', and 8' which intersect each other at points P1, P2, P3, P4, P5, P6, P7, and P8 respectively. See image.
(viii) At last, join these points P1, P2, P3, P4, P5, P6, P7, and P8 by a smooth and thick curve with the help of French curves to get the required Rectangular Hyperbola or Hyperbola by Rectangle Method.
(3) Hyperbola drawn by Parallelogram Method
When a Hyperbola
needs to be drawn through a given point positioned between two lines at any
angle (rather than 90˚) between them, it can be easily drawn with this method.
This method is similar to the Rectangle Method.
(i) First of all, draw two straight lines OA and OB, at a given angle between them (say ɵ).
(ii) Take the point P
as per assumption or specified. See image.
(iii) Now, through point P, again draw two lines in such a way, that CD is parallel to OA and EF is parallel to OB.
(iv) Mark some points 1, 2, 3, 4, and 5 on line PD, and some points 6, 7, and 8 on line PC, as per our assumption or requirements, equidistance is not mandatory.
(v) Now, join O1, O2, O3, O4, and O5 by straight lines which intersect the PF line at points 1', 2', 3', 4', and 5'.
(vi) Similarly, join O6, O7, and O8 by straight lines, and extend those to intersect the line PE at points 6', 7', and 8'.
(vii) Now, draw vertical parallel lines to OB, through the points 1, 2, 3, 4, 5, 6, 7, and 8. And draw horizontal parallel lines to OA, through 1', 2', 3', 4', 5', 6', 7', and 8' which intersect each other at points P1, P2, P3, P4, P5, P6, P7, and P8 respectively. See image.
(viii) At last, join these points P1, P2, P3, P4, P5, P6, P7, and P8 by a smooth and thick curve with the help of French curves to get the required Hyperbola by Parallelogram Method.
What are Asymptots?
When a curve is traced out by a point moving in such a way that the product (multiplication) of its distances from two fixed lines at right angles to each other is a constant, is called a Rectangular Hyperbola (drawn by Rectangle Method, see above). These two fixed lines are called Asymptotes.
Draw a Hyperbola and locate its Asymptots:
(i) First of all, draw a horizontal axis line, and mark the foci F and F1, and vertices V and V1 as per the given dimensions.
(ii) Now, on the axis, mark any number of points 1, 2, or 3 to the right side of F1.
(iii) Take the radius as V-1, and the center as F, and draw two arcs (both sides of the axis line) towards F1. And, take center as F1, draw two arcs (both sides of the axis line) towards F.
(iv) Now, take the radius as V1-1, with the same centers F
& F1, cut those previously drawn four arcs at P1, see image.
(v) Again, take the same centres F & F1, and radius V-2, draw four more arcs
as previous. And, take radius V1-2 to draw arcs intersecting those first four
arcs at points P2.
(vi) Similarly, repeat the process with the same centers and radii V-3 and V1-3 respectively, and draw arcs to get P3. See figure.
(vii) These P1, P2, and P3 points lie on the hyperbola, so, use French curves to draw smooth and tick curves on both sides through these points to get the required Hyperbola.
(viii) Now, take F-F1 as a diameter, and find its middle point as O, and draw a circle.
(ix) Draw two perpendicular lines on the vertices V and
V1, intersecting the
circle at four points A, B, C, and D. See image.
(x) Now, from point O, draw straight lines in a crosswise direction to join all four points. See image. So, these two lines (AOC and BOD) intersecting each other at a right angle (90˚), are the required Asymptotes of the Hyperbola.
How to draw Tangent and Normal on a Hyperbola
(i) Assume a point P through which we’ll draw a Tangent and Normal on the Hyperbola.
(ii) Now, join the focus point F with the point P by a straight line.
(iii) Take the FP line as the base, and draw a right angle (90˚) at point F, in such a way that line FT will intersect the directrix (AB) at point T.
(iv) Join T and P by a straight line to get the Tangent of the Hyperbola at point P.
(v) Now, take the TP line as the base, and draw a right angle (90˚) at point P.
(vi) So, the NPM is the required Normal of the Hyperbola at that point P.