# Drawing a Hyperbola – Engineering Drawing / Technical Drawing

## Drawing a Hyperbola

Definition: It is the locus of a point moving in a plane such a way that the ratio of its distances from a fixed point (focus) to the fixed straight line (directrix) is a constant and is always greater than 1 (i.e. e>1).

The following methods are used to construct a hyperbola :-

- Directrix-focus method
- Rectangular hyperbola

These methods are explained through various examples for drawing a hyperbola. Constructing a hyperbola is very easy using any of the methods.

### Directrix-focus method – Drawing a Hyperbola

Example: A fixed point is 60 mm away from the fixed straight line. Draw the locus of a point P moving in such a way that the ratio of its distance from the fixed point to its distance from the fixed point straight line is 3:2. Name the curve.

### Procedure :-

This method is based on the definition of the hyperbola :-

- Draw any line DD as directrix. Mark any point C on it.
- Draw the axis through C, perpendicular to the directrix DD.
- Mark the focus F on the axis at a distance of 60 mm from C i.e. CF = 60 mm.
- Divide CF into 5 equal divisions.
- Mark the point V (vertex) on CF such that VF/VC = 3/2 = eccentricity = 36 mm/24 mm. Here, eccentricity is greater than 1 and hence asked curve is a hyperbola.
- To construct a scale for ratio 3/2, draw VG = VF at V perpendicular to the axis CF. Join CG and extend it.
- Mark any point 1 on the axis and through it, draw a perpendicular to meet CG produced at 1′.
- With centre F and radius equal to 1 – 1′, draw arcs to intersect the perpendicular through 1 at points P1 and P1′.
- P1 and P1′ are the points on the hyperbola, because the distance of P1 from DD is equal to C-1, P1F = 1 – 1′ and e = P1F/C-1 = 1-1’/ C-1, VG/VC = VF/VC = 3/2.

- Similarly, mark points 2, 3, etc. on the axis and obtain points P2 and P2′, P3 and P3′, etc.
- Draw a smooth curve through points V, P1, P2, … to get the hyperbola. Hyperbola is an open curve.

### Rectangular Hyperbola – Drawing a Hyperbola

Example: A point P is 15 mm and 25 mm respectively from two straight lines which are at right angles to each other. Draw a rectangular hyperbola from P within 5 mm distance from each line.

### Procedure :-

- Draw two axes OA and OB and mark point P as shown.
- Through P, draw lines CD and EF parallel to OA and OB respectively.
- Along PD, mark a number of points 1, 2, 3, 4, etc. at any distances.
- Draw lines O1, O2, O3, O4, etc. cutting line PE at points 1′, 2′, 3′, 4′, etc. respectively.
- Through point 1 and 1′, draw a line parallel to OB & OA respectively, intersecting each other at point P1.
- Similarly, obtain points P2, P3, P4, etc.
- Next, along PC, mark a number of points 5, 6, 7, etc. at any distances.
- Draw lines O5, O6, O7, etc. cutting line PF at points 5′, 6′, 7′ respectively.
- Through point 5, draw a line parallel to OB and through point 5′, draw a line parallel to OA, intersecting each other at point P5.
- Similarly, obtain points P6, P7, etc.
- Draw a smooth curve through all these points P, P1, P2, P3, P4 and P6, P7, etc. which will give a required rectangular hyperbola.

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