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Drawing an Ellipse – Engineering Drawing / Technical Drawing

Drawing an ellipse

Ellipse

 It is the locus of a point which moves in a plane in such a way that the ratio of its distance from a fixed point (focus) to the fixed straight line (directrix) is a constant and is always less than 1. (i.e. e<1)
Or
Ellipse is also defined as a curve traced out by a point moving in the same plane, such that the sum of its distances from two fixed points is always the same.

The following methods are used in Drawing an ellipse:

  1. Directrix-focus method
  2. Arcs of circles method
  3. Concentric circles method
  4. Oblong method
 These methods are explained through various examples on construction of an ellipse. Tangents and Normal at any point on the ellipse are also obtained as described in the following examples.

1. Directrix-Focus method – Drawing an ellipse

Example 1: To draw an ellipse when the distance of the focus from the directrix is equal to 60 mm and eccentricity is 2/3.

Drawing an ellipse 1
 
 

 Procedure:-

 This method is based on the first definition of an ellipse.
  • Draw any line DD as directrix. Mark any point C on it.
  • Draw the axis through point C, perpendicular to the directrix DD.
  • Mark a 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 = 2/3 = eccentricity = 24/36
  • To construct a scale for ratio 2/3, draw VG = VF at V, perpendicular to the axis CF. Join CG & 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 ellipse because the distance of P from DD is equal to C-1.
  • P1F = 1-1’ and e = P1F/C-1 = 1-1’/C-1 = VG/VC = VF/VC = 2/3.
  • 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 an ellipse.

Thus, ellipse is a closed curve and has two foci and two directrices.

Example 2: Draw the locus of a point P which moves in such a way that the ratio ( its distance from a fixed straight line/its distance from a fixed point) is always constant and equal to 5/3. The fixed point is 60 mm away from the fixed straight line. Draw the tangent and normal to the curve at a point on the curve 70 mm from the fixed straight line. Name the curve. 



drawing an ellipse 2

Procedure:-

  • Draw any line DD as directrix. Mark any point C on it.
  • Draw the axis through point C, perpendicular to the directrix DD.
  • Mark a focus F on the axis at a distance of 80 mm from C i.e. CF = 60 mm.
  • Divide CF into 8 equal divisions.
  • Mark the point V (vertex) on CF such that VC/VF = 5/3 i.e. VF/VC = 3/5 = eccentricity = 22.5mm/37.5mm which is less than 1 and hence the asked curve is an ellipse.
  • To construct a scale for ratio 3/5 draw VG = VF at V, perpendicular to the axis CF, Join CG & extend it.
  • Mark any point 1 on the axis and through it, draw a perpendicular to meet CG produced at 1′.
  • With the centre F and radius equal to 1 – 1′, draw arcs to intersect the perpendicular through 1 at points P1 and P1′.
    • P1 and P1′ ar the points on the ellipse, 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 = 3/5.
  • 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…. P1′, P2′ to get an ellipse.
  • Construction of Tangent and Normal to the curve.

Procedure for drawing tangents and normal:-

When a tangent at any point on the curve is produced to meet the directrix, the line joining the focus with this meeting point will be at right angles to the line joining the focus with the point of contact.
 The normal to the curve at any point is perpendicular to the tangent at that point.
 Now, mark a point R on the curve at a point 70 mm from the directrix as shown in the figure. Join FR. Draw a line at an angle 90 degree to FR at F, intersecting directrix at point T. Join TR and extend it, which is the required tangent to the ellipse at point R.
Next, draw MN at R, perpendicular to TR, which represents normal to the ellipse at point R.
Note: A curve parallel to an ellipse and at a distance R from it, ma be obtained vy drawing a number of normals to the ellipse, making them equal to the required distance R and then drawing a smooth curve through their ends.