In this video I go over several new ways of describing curves, which allows for graphing some pretty amazing shapes and drawings, these are: Parametric Curves and the Polar Coordinate system. I go over a brief introduction on these concepts but in later videos I will go into great detail on both topics. Parametric equations involves an additional perimeter, t, to the typical x and y variables, so that they both are functions of t, i.e. x = f(t) and y = g(t)). The resulting curves that can be graphed from these parametric equations are the very same letters and symbols that a laser printer would use. A specific example of the curve that can be graphed from parametric equations is the cycloid, which is a shape formed by tracing a point on the perimeter of a circle as it rotates about a line. This is a pretty cool shape and I briefly go over it in this video.
Polar Coordinates are a special type of coordinate system that is very well suited for circular curves. An example of the type of curve that can be sketched is the Cardioid which is similar to a cycloid except that the circle, which has the point on the perimeter being traced, also rotates about a circle of the same radius. The resulting shape looks like a heart. In fact, polar coordinates can be used to sketch some truly amazing graphs.
I will go into these types of curves in great detail in later video so stay tuned for more!
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Parametric Equations and Polar Coordinates
So far in my videos, I have described plane curves by the following:
- y as a function of x
- x as a function of y
- Implicitly defining y as function of x
In this section, we discuss new methods for describing curves.
Some curves are best handled when both x and y are given in terms of a third variable t called a parameter.
These are called parametric equations and the resulting curves are called parametric curves.
Parametric curves are also used to represent letters and other symbols on laser printers.
An example of such a curve is the cycloid.
https://en.wikipedia.org/wiki/Cycloid
Cycloid
Reveal spoiler
A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage.
Other curves have their most convenient description when we use a new coordinate system, called the polar coordinate system.
An example of such a curve is the cardioid.
https://en.wikipedia.org/wiki/Cardioid
Cardioid
Reveal spoiler
A cardioid (from the Greek καρδία "heart") is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius.
I will go over these new types of describing curves in the videos to come, so stay tuned!
Ahhh, the importance of the pick up pattern! :-)