Search Results for “parabola”
Parabola Polar Line
The locus of the intersection of tangents at the ends of chords of a parabola through a given fixed point is a straight line. This is called the polar line of the given point
Parabola Parametrization
Investigate the characteristics of the standard parametrization of a parabola.
Parabola Subnormal
The subnormal for a given parabola is constant, and closely related to the focal length.
Approximation of a Parabola
The construction is simple: two line segments are each divided uniformly into an equal number of segments. Index the top from left to right, and the bottom from right to left, then connect corresponding points.
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.
Parabola Property
The triangle containing the normal and tangent at a point on a parabola sits inside a semicircle centered at the parabola's focus.
Cubic Spline
Many computer drawing programs use cubic splines to represent arcs. Our diagram shows how a spline can be constructed from 4 control points. We highlight that it is a moving average of two parabolas. These parabolas are exactly the string art parabolas.
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?
Maximum range on a slope
It turns out that (ignoring air resistance) the projectile with the maximum range on a sloping plane follows a parabolic trajectory where the focus of the parabola lies on the plane. Drag the red point to change the slope of the plane. Use the slider to change the muzzle velocity. Can you infer from the properties of a parabola that the initial direction of the projectile bisects the angle between the sloped plane and the vertical?
