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Parametric Department of Mathematics University of Leicester
What is it? A parametric equation is a method of defining a relation using parameters. For example, using the equation: We can use a free parameter, t, setting: and
What is it? We can see that this still satisfies the equation, while defining a relationship between x and y using the free parameter, t.
Why do we use parametric equations Parameterisations can be used to integrate and differentiate equations term wise. You can describe the motion of a particle using a parameterisation: r being placement.
Why do we use parametric equations Now we can use this to differentiate each term to find v, the velocity:
Why do we use parametric equations Parameters can also be used to make differential equations simpler to differentiate. In the case of implicit differentials, we can change a function of x and y into an equation of just t.
Why do we use parametric equations Some equations are far easier to describe in parametric form. Example: a circle around the origin Cartesian form: Parametric form:
How to get Cartesian from parametric Getting the Cartesian equation of a parametric equation is done more by inspection that by a formula. There are a few useful methods that can be used, which are explored in the examples.
How to get Cartesian from parametric Example 1: Let: So that: and
How to get Cartesian from parametric Next set t in terms of y: Now we can substitute t in to the equation of x to eliminate t.
How to get Cartesian from parametric Substituting in t: Which expands to:
How to get Cartesian from parametric Example 2: Let: So that: and
How to get Cartesian from parametric To change this we can see that: And
How to get Cartesian from parametric And as we know that We can see that:
How to get Cartesian from parametric Which equals: This is the Cartesian equation for an ellipse.
Example Example 3: let: Be the Cartesian equation of a circle at the point (a,b). Change this into parametric form.
Example If we set: And: Then we can solve this using the fact that:
Example From this we can see that: So: Therefore:
Example Similarly: So: Therefore:
Example Compiling this, we can see that: Which is the parametric equation for a circle at the point (a,b).
Polar co-ordinates Parametric equations can be used to describe curves in polar co-ordinate form: For example:
Polar co-ordinates Here we can see, that if we set t as the angle, then we can describe x and y in terms of t: Using trigonometry: and
Polar co-ordinates These can be used to change Cartesian equations to parametric equations:
Polar co-ordinates: example Let: Be the equation for a circle. If we set:
Polar co-ordinates: example We can see that if we substitute these in, then the equation still holds: Therefore we can use: As a parameterisation for a circle.
Finding the gradient of a parametric curve To find dy/dx we need to use the chain rule:
How to get Cartesian from parametric: example Example: Let: and Then: and
How to get Cartesian from parametric: example Then, using the chain rule:
Extended parametric example Let: Be the Cartesian equation.
Extended parametric example Then to change this into parametric form, we need to find values of x and y that satisfy the equation. If we set: And:
Extended parametric example Then we have: Which expands to:
Extended parametric example We know that: Therefore we can see that our values of x and y satisfy the equation. Therefore:
Extended parametric example Now, as this is the placement of the particle, we can find the velocity of the particle by differentiating each term:
Extended parametric example Next, we can find the gradient of the curve. Using the formula:
Extended parametric example Using this: And:
Extended parametric example Therefore the gradient is:
Conclusion Parametric equations are about changing equations to just 1 parameter, t. Parametric is used to define equations term wise. We can use the chain rule to find the gradient of a parametric equation.
Conclusion Standard parametric manipulation of polar co- ordinates is: x=rcos(t) Y=rsin(t)