Geometric Construction 5

Construct the center of curvature of an ellipse following these steps:
1. Draw the ellipse according to definition:
2. Draw the straight line connecting the two foci and $F^{\prime }$ :
3. Draw the normal :
4. Draw the straight line connecting the points and $F^{\prime }$ :
5. From the intersection of the lines $FF^{\prime }$ and draw a line parallel to the tangent meeting $PF^{\prime }$ at the point :
6. From draw perpendicular to $PF^{\prime }$ meeting at the required center of curvature. The circle with center passing through is the required circle of curvature:
7. Set up the animation showing various situations:
8. Draw the line segment PQ as P ranges all over the ellipse:
Construct the center of curvature and the circle of curvature of a hyperbola.
Construct the center of curvature and the circle of curvature of a parabola.
Construct the circle of curvature of the cardioid following these steps:
1. Construct the cardioid.
2. Construct the normal.
3. Rotate through by $180^{\circ }$ to the point The required center of curvature is the intersection of the lines and
4. Construct an animation showing how the involute of a cardioid is formed.
Do the same for the nephroid.
Do the same for the deltoid.
Do the same for the astroid.
Construct the conic given four of its points and a tangent passing through one of them.
1. Label the given points as in which the given tangent passes through the point
2. Construct the line meeting the given tangent at the point
3. Construct a ``variable line'' passing through the point and regard it as the Pascal line.
4. Construct the point the intersection of the line $31^{\prime }$ and the Pascal line. Construct the point the intersection of the line and the Pascal line.
5. Locate the required ``sixth point'' $3^{\prime }$ at the intersection of the lines $3^{\prime }1$ and

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