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Drawing an Ellipse 2 – Engineering Drawing / Technical Drawing Arc of Circles Method – Drawing an Ellipse

Example: The major axis of an ellipse is 120 mm long and the minor axis is 80 mm long. Find the foci and draw the ellipse by ‘arcs of circles’ method. Draw a tangent to the ellipse at a point on it 30 mm above major axis. Procedure:-

This method is based on the second definition of an ellipse.
• Draw major axis AB = 120 mm. Mark the midpoint of AB at point O.
• Draw minor axis CD = 80 mm, through o, by considering OC = OD = 40 mm
• Let F1 and F2 be two foci lying on major axisAB. Then F1C + CF2 = major axis = AB or Fc = CF2 = 1/2 major axis = 1/2 * 120 = 60 mm.
• Hence, with C as the centre draw two arcs of radius 60 mm cutting major axis AB at F1 and F2 as shown.
• Between F1 and O take points 1, 2, …, etc. at nearly equal distances.
• Now with F1 and F2 as centres and radii equal to A1 and 1B  draw intersecting arcs on both the sides of AB to get four points P1. Similarly, get points P2, P3, … etc.
• Join above points of intersection by a smooth curve including points A,B,C and D to get an ellipse.
• To get Tangent and Normal at point R.
• Join R with the foci F1 and F2.
• Draw a line NM bisecting ⦟F1 R F2. Then NM is the normal to the ellipse at the point R.
• Draw a line ST through R and perpendicular to NM. Then ST is the tangent to the ellipse at the point R.

Concentric Circles Method – Drawing an Ellipse

Example: The foci of an ellipse are 100 mm apart and the minor axis 70 mm long. Determine the length of the major axis and draw the ellipse by concentric circles method. Procedure:-

• Draw vertical minor axis CD = 70 mm. Mark the midpoint of CD at point O.
• Draw a horizontal line through O.
• Let F1 and F2 be two foci lying on the major axis AB. Now, OF1 = OF2 = 100/2 = 50 mm. By this, mark positions of F1 and F2 as shown in the figure. Then, F1C + CF2 = OB as shown. Measure AB, which is the major axis of the ellipse,
• Observe that major axis AB and minor axis CD intersect each other at O.
• With centre O and diameters AB and CD respectively, draw two circles.
• Divide the major axis circle into a number of equal divisions, say 12 and mark point 1, 2, etc. as shown. Join lines joining these points 1, 2 etc. with the centres O and cutting the minor axis circle at points 1′, 2′ etc.
• Through the point 1 on the major axis circle, draw a line parallel to minor axis CD. Through the point 1′ on the minor axis circle, draw a line parallel to major axis AB. Intersection of these two lines is a point P1, which lies on the required ellipse.
• Similarly, considering the points 2, 3, etc. on the major axis circle and 2′, 3′ .. etc. on the minor axis circle respectively, we can obtain points P2, P3 .. etc. which are lying on the required ellipse.
• Draw the smooth curve passing through all the points A, P1, P2, … etc. which will give the required ellipse.

Oblong Method – Drawing an Ellipse

Example: The major axis and minor axis of the ellipse are 125 mm and 75 mm respectively. Construct an ellipse by oblong method. Procedure :-

• Draw rectangle EFGH with larger side as major axis of 125 mm length and smaller side as minor axis of 75 mm length.
• Draw line AB and CD as centre lines of the rectangle intersecting at O as major axis and minor axis respectively.
• Divide the semi-major axis AO into a number of equal parts, say 5, and AE into the same number of equal parts, marking them from A as shown.
• Draw lines joining 1′, 2′, 3′ and 4′ with C.
• From D, draw line through 1, 2, 3 and 4 intersecting C1′, C2′, C3′, and C4′ at points P1, P2, P3 and P4 respectively.
• Do similar constructions for all four parts of rectangle and get points P1, …., P4, etc. In all, we get 5*4 = 20 points. Join all these points by a smooth curve to get an ellipse.