MHB What is the difference between a linear spline and a linear B-spline?

[mathmari]

Hey!

We want to construct a linear spline that at the points $x_0=-1$, $x_1=0$, $x_2=1$ has the values $b_0=0$, $b_1=1$ and $b_2=3$.

The spline should get from $S_{x,1}$ with $x=\{x_0, x_1, x_2\}$ to $[-1,1]$.

Do we want to find a function of the following form:

$$s(x)=\begin{cases}a_1x+b_1 & x\in [x_0, x_1] \\ a_2x+b_2 & x\in [x_1, x_2]\end{cases}$$
that satisfies the given values? (Wondering)
B-splines belong to a specific category of splines, right? What are the differences between the B-splines and other splines? (Wondering)

 

[I like Serena]

Hey!

We want to construct a linear spline that at the points $x_0=-1$, $x_1=0$, $x_2=1$ has the values $b_0=0$, $b_1=1$ and $b_2=3$.
The spline should get from $S_{x,1}$ with $x=\{x_0, x_1, x_2\}$ to $[-1,1]$.

Do we want to find a function of the following form:
$$s(x)=\begin{cases}a_1x+b_1 & x\in [x_0, x_1] \\ a_2x+b_2 & x\in [x_1, x_2]\end{cases}$$
that satisfies the given values?

Hey mathmari!

Yep. (Nod)

B-splines belong to a specific category of splines, right? What are the differences between the B-splines and other splines? (Wondering)

From wiki:

In mathematics, a spline is a special function defined piecewise by polynomials.​

and:

In the mathematical subfield of numerical analysis, a B-spline, or basis spline, is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree.​

(Nerd)

To be fair, I originally learned that splines are indeed piecewise functions defined by polynomials.
And usually those polynomials are of the order 3 with 1st and 2nd order continuity at the knots.
The main families of 3rd order splines with 1st and 2nd order continuity that I learned were:

• Hermite spline: defined to pass exactly through the control points. Each piece is defined by 4 control points.
• Bézier spline: defined to pass through control points with specific tangent vectors at those control points. Each piece is define by 2 control points and 2 tangents.
• B-Spline: defined to pass through first and last point, and otherwise pass between the control points with a tangent corresponding to the neighboring points. Each piece is defined by 4 control points.

(Thinking)

 

[mathmari]

So we get the linear spline function: \begin{equation*}s(x)=\begin{cases}p_1(x)=x+1 & x\in [-1, 0] \\ p_2(x)=2x+1 & x\in [0, 1]\end{cases}\end{equation*}

If we want to construct a linear B-spline, would we have to do something else?

Or do they have the same form as a linear spline but just have different properties?

(Wondering)

From wiki:

In mathematics, a spline is a special function defined piecewise by polynomials.​

and:

In the mathematical subfield of numerical analysis, a B-spline, or basis spline, is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree.​

(Nerd)

To be fair, I originally learned that splines are indeed piecewise functions defined by polynomials.
And usually those polynomials are of the order 3 with 1st and 2nd order continuity at the knots.
The main families of 3rd order splines with 1st and 2nd order continuity that I learned were:

• Hermite spline: defined to pass exactly through the control points. Each piece is defined by 4 control points.
• Bézier spline: defined to pass through control points with specific tangent vectors at those control points. Each piece is define by 2 control points and 2 tangents.
• B-Spline: defined to pass through first and last point, and otherwise pass between the control points with a tangent corresponding to the neighboring points. Each piece is defined by 4 control points.

(Thinking)

What exatcly does "pass through first and last point" mean? (Wondering)

 

[I like Serena]

So we get the linear spline function: \begin{equation*}s(x)=\begin{cases}p_1(x)=x+1 & x\in [-1, 0] \\ p_2(x)=2x+1 & x\in [0, 1]\end{cases}\end{equation*}

If we want to construct a linear B-spline, would we have to do something else?

Or do they have the same form as a linear spline but just have different properties?

For a linear spline there's not really a choice.
We have insufficient parameters to control the 1st and 2nd order derivatives.
So I think a linear spline function and a linear B-spline may indicate the same function. (Thinking)

What exatcly does "pass through first and last point" mean?

That it looks like this:
View attachment 7702
In this example the B-spline starts at the first point and ends at the last point.
The B-spline does not pass through any of the other points. (Thinking)