Understanding Cardinal Splines - Formula Explained

[Gepard]

Hi,

I'm trying to draw a Cardinal Spline and understand them - however I can't get my head around the formula.

Now, after lots of research I keep coming back to this formula:

Ti = a * (Pi+1 - Pi-1 )

taken from this page(there's a similar one on Wikipedia): http://cubic.org/docs/hermite.htm

However I don't understand how that formula can provide a tangent. In the past all the formulae I have used have been of function types where I'd put through say the X co-ordinate and it would tell me the Y co-ordinate.

Can someone please provide a better explanation as to how I use it please?

Thanks in advance,

Michael

 

[hotvette]

I think maybe you are looking for a physical interpretation of the formula and how that relates to the slope at the free end. It might help to break P into its' x & y components:

T_ix = a * ( x_i+1 - x_i-1)
T_iy = a * ( y_i+1 - y_i-1)

In a parametric curve, the slope of the curve is dy/dx = (dy/ds) / (dx/ds). The formulas above represent (dy/ds) and (dx/ds). I hope this helps.

With splines, the end points are arbitrarily chosen by the person implementing them. I can freely choose to say T1 = -42 and T2 = 0.4 and have a perfectly valid spline. It doesn't really matter how I derived them. In the case of cardinal splines, they provide a special mechanism (a control point and tension parameter) as a convenient way to specify the end condition.

 

[tacman]

Hi Michael,

I completely understand your confusion with the formula for Cardinal Splines. The reason it may be difficult to understand is because it is not a typical function type formula, like you mentioned. Instead, it is a formula that calculates the tangent vector for a point on the spline curve.

To better understand this, let's break down the formula into its components. First, let's look at the "a" term. This represents the tension of the spline, which essentially controls how smooth or sharp the curve will be at that particular point. A higher tension value will result in a sharper curve, while a lower tension value will result in a smoother curve. This allows for more control and customization of the spline curve.

Next, we have the "Pi+1" and "Pi-1" terms. These represent the points that come before and after the point we are calculating the tangent for. For example, if we are calculating the tangent for point Pi, then Pi+1 would be the next point on the curve and Pi-1 would be the previous point. These points are important because they help determine the direction of the tangent vector.

Now, let's look at the overall formula again. The result of this formula is a vector, which has both a magnitude and direction. The magnitude is determined by the "a" term, which we already discussed. The direction is determined by the difference between Pi+1 and Pi-1. Essentially, this formula is taking the difference between these two points and scaling it by the tension value to get the final tangent vector.

I hope this explanation helps in understanding the formula for Cardinal Splines. Remember, it is not a typical function type formula, but rather a formula for calculating the tangent vector at a specific point on the spline curve.