Search Results for “envelope”
Circle Envelope (Hyperbola)
The hyperbola has foci A and B. The family of circles whose center C lies on the hyperbola and which pass through the fixed point D defines an envelope curve. What does this envelope look like when D lies on A or B?
Circle Envelope
The ellipse has foci A and B. The family of circles whose center C lies on an ellipse and which pass through a fixed point D defines an envelope curve. (Press Show). What does the envelope look like when D lies on one of the foci of the ellipse.
parabola envelope
We use a trick to let the trace "open up" as you drag a point. The trick is this: an initial point is given parametric location s*t, create a tangent at this point and its envelope as s varies. Now hide the original point and create another point with parameter t, and make it draggable. Dragging the new point changes the value of t and we see a trace from 0 to t.
And the envelope please...
What is the phantom curve you see when you look at a set of lines, perpendicular to a set of chords through a common point in a circle?
Circle isotomic
The circle isotomic is the locus of the reflections of a given point in the tangents to the circle. It is easy to convince yourself that it is also the envelope of the circles whose centers lie on the circle and which pass through the given point. The given point is C. Try dragging it outside the circle.
Simpson Line
Point E lies on the circumcircle of triangle ABC. The projections of E onto the sides of the triangle are colinear. The envelope of such lines forms a deltoid curve. Try dragging E round the circle. Try dragging A B and C. Note that the envelope curve appears to remain equilateral.
Caustic of y=f(X)
The thick red curve is y=f(x), initially f(x) = cosh(x). The thinner red curve is the caustic formed by light parallel to the y-axis reflecting in y=f(X). (The caustic is the envelope of the reflected rays. The envelope of a family of lines is the curve which is tangential to the entire family). The blue curve is the parabola which is the 2nd order Taylor expansion of f(X) about the point J I is the focus of this parabola. What do you notice about Ias you drag J?
Caustic Formation
Light from C reflects in the tangent to the circle at point B. Observe the trace as B rotates around the circle. While the individual rays move, the phantom curve - the caustic - does not. The caustic is the mathematical envelope of the reflected rays, which is to say that each reflected ray is tangential to the envelope curve. At any moment, each ray is moving with a combination of rotation and translation. Each point on the ray is undergoing a translation. Points on the ray which lie on the caustic are moving along teh length of the ray, and hence their motion is not visible. Which is why the caustic looks stationary while everything else moves.
Coffee Cup Caustic
You can move the light source and observe the changes in the reflected rays which form the caustic.
