Finding the curvature of a trig function is a tricky business. Most times, you cannot just find the curve at the point!
Thankfully, for this article, we will use the trick of using matrix inversion. Let’s get started!
Matrix inversion is a technique that has been used for years tofindthecurvatureofatrigfunction.Youcanreadaboutithere,butfornow,let’sjustadvancedmatrixinversionthatfindsthecurvatureat(7,1,1).
That is why we will use matrix inversion to do so! Here are some tips on how to do it.
Calculate the point-value of r(t) at (7, 1, 1)
If you want to know the point-value of r(t) at (7, 1, 1) in T3, you can calculate it as follows.
Take the difference between r(t) and t and add 1.
Subtract this from 7 and you have your value of r(t).
This is called a Riemann-zeta function, and it has a very rare point-value. Only a few thousand copies of this function exist in the world, making this an extremely valuable function to find.
This function is not found in any books or online sources, making it a difficult curve to find.
Calculate the curvature using the formula for curvature
The curvature of a curve can be calculated using the formula for curvature. This is done by taking the sines of the curve and then finding the point where the sine changes into a straight line.
The term sin is used to find the sine of a number, so when finding the point on a curved line, this number must be small. For example, on a normal (or constant) line, 1/1 would be a simple sin(1/1) = 0, so we can simply write 0 for this number.
The formula for curvature is: c = P / (2π π / π), where c is the curvature and P is the scale factor. So, c = F / (2π π / π), where f is the scale factor.
Make a table of values for r(t) and its derivatives at (7, 1, 1), and calculate the curvature each time
When you know the value of r(t) for a certain time interval t, you can calculate the derivative of r at any t-value.
For example, if r = 7t, then the derivative of 7 is 1/7, and if 7t = 14, then 1/7 + 1/14 = 3/14, which is 4.
As mentioned earlier, T2 and T3 are special points on the curve where r(t) equals 4 times its value at those points. These are called quadrature points, because they give you a square root of your curve when you put in those values.
By definition, a function whose value depends on one or more other values is defined to be curvature-dependent.
Graph r(t), and find its maximums and minimums along with their corresponding values of r(t) and its derivatives at those points
In the case of the constant 7, finding the curve at the point (7, 1, 1) means solving for r(t) along t.
For t > 0, you want to find r(0), which is a constant value of 4. Similarly, for t negative value of 4.
To solve for r(t), you need to find a function f that maps into t − 4 and that has a value of 4 for t − 2 and 4 for t + 2. The equivalent equation to this function is f(4).
Compare your results with those from your table to check your work
If your table shows that there is an exact match at t = 1, then the two equations that determine the solution curve at t = 1 can be found in the table’s cells by clicking on the corresponding hyperlink.
The equations are: r(1) = 7t + t2, and r(1) = 7t − 1. So, if your first solution is equal to 7, then you can add or subtract one of these two values to find another solution with a different curvature.
Likewise, if your first solution has a curvature of −1, then you can multiply or divide one of these two values by −1 to find another with a different curvature.
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