# Search Results for “morph-circle-into-line”

##### 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##### Polar Line

The locus of the intersections of the tangents at the end of chords of an ellipse which pass through a common point is a straight line. A bit of a mouthful, but play with the diagram and it will become clear.##### Hyperbola Polar Line

The polar line is the locus of the intersections of tangent lines at the ends of chords of teh parabola through a fixed point. Turns out to be conceptually important - not just a curiosity.##### 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.##### Basic Unit Circle

This very basic representation of the unit circle displays the unit circle with an input for the standard angle θ in degrees (which controls the angle between the hypotenuse and the x axis). The outputs represent the other two sides of the triangle and give their lengths through decimals. A good investigation for geometry students is to have them test out different angles here, then compare the results to those testing the angles with sine and cosine on their calculators. This allows them to visualize the unit circle in a precise diagram rather than simply running inputs and outputs on their calculators.##### 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.

##### Polar Proportional Point and Circle Puzzler 3

A circle is centered at the origin and its radius is defined by the distance between the origin and a point, P. P is defined by a polar function, u(*t*), and is located at the current value of

*t*. Adjust

*t*, or animate it by pressing ”go”. What is u(

*t*)? Look at the .gx source for the answer.

*Hint: if you look closely, u(t) can be seen in dark gold. It is a particular ”polar flower”.*

##### 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.*

##### Polar Proportional Point and Circle Puzzler 2

A circle is centered at the origin and its radius is defined by the distance between the origin and a point, P. P is defined by a polar function, s(*t*), and is located at the current value of

*t*. Adjust

*t*, or animate it by pressing ”go”. What is s(

*t*)? Look at the .gx source for the answer.

*Hint: if you look closely, s(t) can be seen in dark purple. It is a particular ”polar flower”.*