Curves (conic sections): a solved example
Construct an ellipse whose foci are 60 mm apart and whose major axis is 100 mm long. Also draw the tangent and normal to the ellipse at any suitable point.
Every line below was drawn and checked by Graspline's own geometry engine, by the textbook method, one step at a time with the reason for each.
The construction, step by step
What the question gives
Draw the major and minor axes, meeting at the centre O.
Locate the foci, 30 mm either side of O.
Divide and construct
Divide the segment between the foci into 12 equal parts. Through each division point strike two arcs, struck from the foci, of radii 50 mm ∓ its distance from O. The two radii always add to the major axis, so every crossing is a point of the curve.
The curve
Join the constructed points with a smooth curve. It is a ellipse, so every ordinate is a chord of it..
Tangent and normal
At P — 20.8 mm from O along the axis and 36.4 mm off it — the tangent cuts the axis where the product of the two distances is a² = 2500, and the normal is drawn at right angles to the tangent at P.
Result
A ellipse of semi-axis 50 mm and 40 mm, drawn by the arcs method from 13 constructed points, with the tangent and normal at P.
✓ Verified: every constructed point lies within 0.000001 mm of the curve, and the tangent is at right angles to the normal direction there.
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