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By Nick Halsey
Epic Circle Trace 3
Four points are located proportionally around a circle, according to four different functions of t. A figure connecting the four points is traced through t. What are the four functions? Look at the .gx source for the answer. Tip: press "go" to animate t at a constant rate from 0 to ∏ and back, looped.

Tags: Proportional-point, function, circle, puzzler, trace, ellipse, sometimes-quadrilateral

By Nick Halsey
Epic Circle Trace 2
A triangle, defined by three points that are located proportionally around a circle by functions of t, is traced as t varies from 0 to 2Π. What are the functions of t, f(t), g(t), and h(t), that define the points D, E, and F, respectively? Hint: one of the functions is _(t) = t.

Tags: Trace, puzzler, circle, triangle, proportional-point, functions

By Nick Halsey
Epic Circle Trace
A line passes intersects a circle at two points. Each point is located proportionally around the circle in terms of a given function of t. The path of the line’s movement is traced as t varies. Try changing/animating t. Can you figure out how each point is constrained, in terms of t? Look at the gx source file for the answer. Hint: look at the period of the movement, and how it changes as t changes.

Tags: Trace, puzzler, circle, proportional-points, functions

By Nick Halsey
Two Circle Trace
This trace follows the position of one circle as it moves along the path of the edge of another.

Tags: Circles, Traces

By Phil Todd
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.  

Tags: parabola, envelope

By Phil Todd
A model of a pantograph:  what you trace with the red point is replicated at half scale at the blue point.

Tags: mechanism, pantograph, linkage

By Phil Todd
Train Problem
Train AB leaves the station at a constant speed, while train CD approaches the station at a different constant speed.  We trace the locus of the intersection of the lines joining the fronts of the trains and their backs. You can drag B to see the trains move, change their lengths and relative speeds.

Tags: conic, train

By Phil Todd
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.

Tags: caustic, reflection

By Duncan
Rolling Parabola
If you roll a parabola along the ground, what curve does its focus trace out? It might look like a parabola, but is it?

Tags: parabola, catenary, focus

By Nick Halsey
Geometric Top View Model of the Twisted Savonius Wind Turbine (Interactive)
This app models the top view of the Twisted Savonius Vertical Axis Wind Turbine (VAWT). The various inputs and draggable points allow you to see how the model can trace the blades' surfaces. You can also control the twist angle, radius, and rotation - which makes the whole thing spin! Learn more about the Twisted Savonius Modeling Project here.

Tags: Twisted-Savonius, Top-View, Model, Geometric, Real-World, Circles, Arcs, Loci

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