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This essential trains for: AMC-8, AMC-10, AMC-12, AIME, Math Kangaroo 9-10.
What is the difference between an ellipse and an oval?
The ellipse is a conic curve, which we discuss in another lesson. It has its own definition as a locus.
The oval is a shape produced by smoothly connecting several arcs of circle.
This lesson is about producing ovals and other smoothly connected curves.
By smooth connection we mean that two curves have the same tangent line at the point where they are joined (plus an additional condition related to the "direction" or the join that we will get to in a moment).
If two joined curves have different tangent lines at the junction point then they are said to form a kink. We are not interested in such curves in this lesson.
The radius of a circle is perpendicular to the tangent line at the point of contact. If two circular arcs are smoothly joined then they share a tangent and they also have radii that are segments of the same line.
The same graphical conditions are fulfilled if the two arcs form a cusp, i.e. a point where the derivative changes sign:
Or if two arcs are tangent interior:
Every point on a smooth curve has a tangent line. If a smooth curve is assigned a "direction," then the tangent line at each point can be specialized to a tangent ray pointing in the correct direction. A connection of the ends of two curves is "smooth" if it is possible to assign compatible directions to each curve from the join point such that the endpoints of the curves have the same tangent ray. A pair of direction assignments is "compatible" if one direction is "towards" the join point while the other direction is "away" from the join point (rather than having both point "towards" the join point or both point "away" from the join point).
Ovals are constructed using 4 arcs of circles that are joined smoothly. Let us first draw half an oval:
Thus two circles of different radii have been joined smoothly to form half of an oval. The other half is formed analogously.
When solving problems related to ovals, the actions must be to: