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Drawing a Parabola – Engineering Drawing / Technical Drawing

parabola

Parabola

Definition : It is the locus of the point moving in a plane in 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 equal to 1. (i.e. e=1).
There are 3 method using which a parabola is constructed :-
  1. Directrix – focus method
  2. Rectangle method
  3. Tangent method
These 3 methods are used through various examples for drawing a parabola
Tangents and normal at any point on the parabola are also obtained as described in these examples.

Directrix-Focus Method – Drawing a Parabola

Example: Construct a parabola, when the distance of the focus from the directrix is 60 mm and draw a tangent and normal at any point on it 45 mm from F.
 Drawing a parabola 1

 

Procedure :-

  • Draw any line DD as directrix. Mark any point C on it.
  • Now draw the axis through C, perpendicular to the directrix DD.
  • Now mark a focus F on the axis at a distance of 60 mm from C i.e. CF = 60 mm.
  • Divide the ine CF into six equal divisions.
  • Mark the point V (which is the vertex) on CF such that, VF/VC = 1 = eccentricity = 30 mm/30 mm.
  • To construct a scale for ratio 1, 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 parabola, because the distance of P1 from DD is equal to C-1, P1F = 1-1′ and e = P1-F/C-1 = 1-1’/C-1 = VG/VC = VF/VC = 1.
  • Note: For e=1, the line CG makes an angle of 45 degree with the axis of the parabola and hence P1 and P1′ points are also obtained with F as centre and radius = C-1 and drawing arcs cutting the perpendicular through 1 at P1 and P1′.
  • 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 parabola.
          Thus, parabola is an open curve.
  • Now to get tangent and normal at a point R on the parabola.
    • Mark R on the parabola FR = 45 mm
    • Join RF
    • Draw a line FT through F and perpendicular to FR. Then TR is the tangent to the parabola at point R. Extend TR.
    • Draw line MN, perpendicular to TR at the point R, which will represent the normal to the parabola at point R.

Rectangle Method – Drawing a Parabola

Example: The throw of a ball from a fielder on a cricket ground reaches the wicket-keeper’s gloves, following parabolic path. Maximum height achieved by the ball above the ground is 31 m. Assuming the point of throw and the point of catch to be 1 m above the ground. Draw the path of the ball. The radial distance between the fielder and the wicket-keeper is 75 m.
Drawing a Parabola 2

Procedure :-

                When the base and axis of the parabola are known, then rectangle method is used to construct the parabola.
                In this problem, the axis length is the maximum height achieved by the ball and which is 31 m and the base length is the radial distance between the fielder and the wicket-keeper and which is 75 m. Therefore, AB = 75 m and CD = 31 m.
  • Draw a line AB which is the base of length 75 m.
  • Locate midpoint of AB as point D.
  • Draw the axis CD of  31 m length and at right angles to the base AB.
  • Construct a rectangle ABFE as shown.
  • Divide AD and AE into the same number of equal parts, say 5, and name them.
  • Draw lines joining C with points 1′, 2′, 3′ and 4′.
  • Through 1, 2, 3 and 4 draw perpendiculars to AB intersecting C1′, C2′, C3′ and C4′ at points P1, P2, P3 and P4 respectively.
  • Draw a curve through A, P1, P2, P3, P4 and C which is a half parabola.
  • Repeat the same procedure in order ti construct the other half of the parabola.
          Thus, the complete parabola can be obtained which represents the parabolic path followed by a ball as shown.

Tangent Method – Drawing a Parabola

Example: To construct a parabola with a base of 80 mm and axis height of 40 mm by the tangent method.

Procedure :-

  • Draw the line AB which is the base of 80 mm length.
  • locate midpoint of AB and denote it by D.
  • Draw the axis CD of 40 mm length and at right angles to the base AB.
  • Produce DC to 0 so that DC = CO.
  • Join OA and OB.
  • Divide lines OA and OB into the same number of equal parts, say 7. Name the division points as shown in the figure.
  • Draw lines joining 1 with 1′, 2 with 2′, 3 with 3′, etc.
  • Draw a curve starting from A and tangent to lines 1 – 1′, 2 – 2′, 3 – 3′, etc. and ending at B.

         The curved obtained, is the required parabola.Tags: parabola, parabola in engineering drawing, constructing a parabola, different types of construction of parabola, 3 ways of drawing a parabola. different types of making a parabola, parabola using directrix-focus method, parabola using rectangle method, parabola using tangent method.