| 1. | The formula follows by verifying it for the osculating circle.
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| 2. | This is the osculating circle to the curve.
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| 3. | The "'center of curvature "'is the center of the osculating circle.
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| 4. | The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.
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| 5. | This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface.
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| 6. | The curve " C " may be obtained as the envelope of the one-parameter family of its osculating circles.
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| 7. | Each constraint can be a point, angle, or curvature ( which is the reciprocal of the radius of an osculating circle ).
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| 8. | If " P " is a vertex then " C " and its osculating circle have contact of order at least four.
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| 9. | Given any curve and a point on it, there is a unique circle or line which most closely approximates the curve near, the osculating circle at.
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| 10. | :I'm not well-versed on the mathematics involved, but you might be able to construct the parabola by treating the log as an osculating circle.
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