[color=#0000ff][b][size=100][size=200][size=150]The osculating circle to a curve in a given point is the circle which best approximates the curve around the point. In the app below, you can see a parabola and its osculating circles. [br][br]Exercice 1: Move the slider at the top of the applet to move the point P along the parabola and watch the behaviour of the osculating circle. Try to relate the radius of the circle to the shape of the curve around P. [/size][/size][/size][/b][/color]

[color=#0000ff][b][size=150]The curvature of a curve at a point P measures the closeness of the curve around the point. [br][br]Exercice 2: Look at the values of the curvature of the parabola at the moving point P. Try to relate these values with the radii of the corresponding osculating circles.[/size][/b][/color]

[size=150][color=#0000ff][b]Exercice 3: Analyze the behaviour of the osculating point to the following curve at its flex points. What is the value of the curvature at these point?[/b][/color][/size]

[b][color=#0000ff][size=150]Exercice 4: Change the definition of the functions X(t), Y(t) and the extremes tmin and tmax at the top left of the applet to study the osculating circle and the curvature on other curves. Try at least a circle and an ellipse.[/size][/color][/b]

[b][color=#0000ff][size=150]Exercice 5: Describe the curvature of a circle[br][br]Exercice 6: What are the maximum and minimum values of the curvature on an ellipse?[/size][/color][/b]