Parametric Equations and Polar Coordinates. Parametrizations of Plane Curves

Parametric Equations and Polar Coordinates.

Parametrizations of Plane Curves

In mathematics, parametrization is the process of assigning a representation to geometric objects through their coordinates, a way of describing the curve in a coordinate system. This can refer to the parametrization of curves in the plane, in space or a higher dimensional space.

A cycloid is a curve traced by a point on the rim of a circular wheel as it rolls along a straight line. The cycloid has been called the "most complex curve known to mathematicians" because of its analytical complexity, and for good reason: it can be used to solve many difficult problems in both physics and mathematics.

Slope is the rise over run. The rise is the vertical distance between two points on a line, and the run is the horizontal distance between them. A line has a definite slope when it has a definite direction.

Part 1. Parametrization of plane curves

One approach to graphing a function is to write y as a function of x, so that we can draw the graph in the x-y plane by using such an equation.

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There are indeed some situations in which it's more convenient and natural to describe the relationship between x and y in other ways.

In the following example, let's imagine that there is a small animal that moves along the curve of a student. It's very natural to describe the location of the animal by plotting its (x,y) coordinates at each time point t on a graph, so for example, along this graph, this point here denotes the (x,y) coordinates of the animal at time t.

The relation between the x coordinate and the y coordinate is through a third party, which we call t. So, this is a situation in which it's much more natural to describe locational points through a third party which we call parameter. More precisely, parametric equations are defined as follows.

Parametric equations and parametric curves describe a pair of functions that depend on a single variable. The two functions are usually written in terms of a parameter t (x equals to a function g of t and y equals the function f of t).

We also call this the parameter, which ranges over an interval. If we call the parameter t, then the set of points with coordinates x and y which is equal to g(t) and f(t) defines a parametric curve like this blue curve here and this pair of equations, x equal to g(t), y equals to f(t) are called parametric equations for the curve.

If a function f(t) and a variable ft are defined over an interval, then the set of points (x,y)=(g(t),f(t)) defines a parametric curve. These equations are called parametric equations for the curve.

To find a parametrization for the straight line with slope m and intercept c, first consider the equation for such a straight line given in rectangular coordinates: y = mx + c.

Let us set t equal to x. Then the equation becomes y equals mx plus c. Therefore, the parametric equations for the straight line are: x(t) equals t and y(t) equals mx plus c where t ranges from minus infinity to infinity.

The above pair of parametric equations may also be expressed in vector form. The vector is two by one, the first component x equals to zero plus t times one, so it is x equals to t; the second component y equals c plus mx, so this recovers the same pair of parametric equations.

Example: Find a parametrization for the straight line with slope m and y-intercept c.

This next example is a parametric curve given by the pair of equations, x equals cosine t, y equals sine t, where t goes from zero to two pi.

The points of the parametric curve are depicted in this circle:the parameter theta is shown here.When theta equals zero, it corresponds to this point; as theta increases, it corresponds to points on this circle going around entire clockwisely. When theta is equal to pi/2, the point is here; when theta equals pi/2, it corresponds to a point minus one (zero here); as theta increases further, this point again moves around this circle in this way.

So it can be seen that this pair of equations describes this blue circle.

Next, consider the following pair of parametric equations: x equals to the square root of t, and y equals two t. Now, in this case, t must be greater than or equal to zero because otherwise the square root of t will be undefined for negative values.

To plot this graph, we make a table with three columns: t, x, and y. For each value of t, we compute x and y accordingly so that we have columns against each value of t. This gives us our parametric curve. The point (0,0) corresponds to t=0 and (1,1) corresponds to t=1.

This is the point for t equals to four. After putting these points, the curve given by this pair of parametric equations is the solid blue line in the figure.

Next one.

The parametric equations for this pair of curves are x equals the square root of t, y equals two t for t greater than or equal to zero. In this case, t has to be greater than zero because otherwise the square root of t will be undefined for negative values.

To plot this graph, we make a table with three columns, t, x and y. The values under t are increasing from zero to some maximum value and we compute x and y accordingly. We plot these points on a coordinate plane, forming a curve known as a parametric curve. The point (0,0) corresponds to t equal to zero, while the point (1,1) corresponds to t equal one.

This is the point where t equals four. After graphing these points, the curve given by this pair of parametric equations is shown by this solid blue line. The next example is x equals t, y equals two t squared.

We first make a table consisting of the three columns, t, x and y. In this case, t can range from minus infinity to t because we do not have the restriction that t must be non-negative; we do not have square root here.