b-spline | B-spline interpolation | Game Engine library

I am checking out the mgcv package in R and I would like to know how to update a model based on new data. For example, suppose I have the following data and I am interested in fitting a cubic regression spline.

I have 2 function to either calculate a point on a spline, quadratic or cubic:

In the r programming language, the following

In cases like this is, is it possible to display a smooth curve by smoothing the levels where values drop?

For example, here are two complete lists.

In the documentation for geom_smooth(), there is an example that shows how to fit a B-spline smooth to the hwy vs. displ columns of the tidyverse mpg dataset, using a parameter setting for the bs() function of df=3:

I'd like to repeat the same example, but instead of computing just a single smooth with a single setting for the df parameter, I'd like to use a range of df values (for example, 3, 5, 7, 9) to calculate a series of smooths, and then display each smooth in a separate panel using facet_wrap() (and also as a minor addition, I furthermore want to display the gray-shaded confidence interval around the smooth curve). However, I can't quite figure out what syntax I should use, or indeed whether ggplot2 even has the flexibility to support a computation such as this directly inside of geom_smooth().

I've posted a MWE below:

I'm working on saturate B-spline basis using standard roughness penalty.

I'd like to have plots like below:

But my plot looks like this:

I don't know why my plot() function cannot change the color of lines. Here is my code below and could you please tell me the reason with solution? Thank you in advance!

I am wondering how you would model arbitrarily complex Bézier curves. I am not quite understanding the underlying abstraction yet of what a bezier curve is fundamentally composed of, as there are too many equations. I would like to have a generic struct that defines a bezier curve. The SVG path gives many examples of the types of curves you can create. They include linear, cubic, and quadratic bezier curves.

If a B-spline is a better generic model, then that would be fine to use too. I am not familiar with those yet tho. Difference between bezier segment and b-spline. I guess "a B-spline curve is a curve that consists of Bezier curves as segments", so that is what I am looking for.

SVG docs say:

Cubic Béziers take in two control points for each point.

Several Bézier curves can be stringed together to create extended, smooth shapes. Often, the control point on one side of a point will be a reflection of the control point used on the other side to keep the slope constant. In this case, a shortcut version of the cubic Bézier can be used, designated by the command S (or s).

The other type of Bézier curve, the quadratic curve called with Q, is actually a simpler curve than the cubic one. It requires one control point which determines the slope of the curve at both the start point and the end point. It takes two parameters: the control point and the end point of the curve.

Arcs and NURBS (non-uniform rational B-splines) are more complex than just plain bezier curves, but it would be nice if the model could be generalized enough to include these as well. Basically I would like a generic model of bezier curves/b-splines/nurbs to use in a drawing/graphics framework, and not sure what that would be.

• Must each bezier class be implemented separately, or can they be combined into one generic class?
• If separate, are they each basically just an array of control points?

So basically I start to think:

The content and pictures on Wikipedia indicate that the B-spline does not pass through the control point, and the information I found elsewhere is the same as the content on Wikipedia. However, spline interpolation can be performed in scipy.interpolate, such as interp1d 'cubic' or splrep splev(the two operations should be similar Difference between quadratic and 2nd order spline interpolation in scipy). Through drawing, I found that the curve passed all control points. But the documentation and library code show that the basis of these operations is B-spline.

I am deeply confused about this now. I know that B-spline is a kind of spline, and there are many kinds of spline interpolation besides B-spline interpolation, such as natural cubic spline interpolation and cubic Hermitian interpolation.

My current problem:

1. The relationship between spline curve and spline interpolation. Does the spline interpolation always pass through all control points?
2. Are B-spline and B-spline interpolation and cubic spline interpolation the same?
3. Will B-spline interpolation pass through control points?

My understanding is that rcs() (from the rms package) uses a truncated-power basis to represent natural (restricted) cubic splines. Alternatively, I could use ns() (from the splines package) that uses a B-spline basis.

However, I noticed that the training fits and testing predictions could be very different (especially when x is extrapolated). I'm trying to understand the differences between rcs() and ns() and whether I could use the functions interchangeably.

Fake non-linear data.