# What Does the F Matrix Look Like for a Linear Bezier Curve?

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• bobtedbob
In summary, the conversation discusses the mathematics behind rendering bezier curves and specifically focuses on the creation of the F matrix for the linear bezier case. The paper referenced in the conversation provides information on the calculation of the M matrix and its inverse, but the process for creating the F matrix is unclear. The moderator notes that the F matrix is a permutation matrix and provides an explanation for its form. They also suggest searching for additional resources on Bézier curves for further understanding.
bobtedbob
TL;DR Summary
Rational linear parametric curve and its implicit
form that is a projected image of the algebraic curve
I'm looking at the following web page which looks at rendering bezier curves.

GPU Gems 3 - Chapter 25
Paper on same topic

The mathematics is quite interesting, I was interested to know what the F matrix would look like for for a linear bezier equation. The maths for the quadratic case is in the paper (2nd link) section 3 claim 1. I understand how the M matrix is calculated (and its inverse) but I don't understand how the F matrix was created.

Can someone help explain the F matrix creation process and how it would apply to the linear bezier case?

[Moderator's note: approved.]

Last edited by a moderator:
In paper (2nd link) section 3 claim 1, the matrix F is a permutation matrix. This matrix F has the given form because the vector ##v=[1\,\, t\,\, t^2]## was rewritten as ##u=[t \,\,t^2\,\, 1]##. This is represented by the matrix F, which is the identity matrix, rewritten with colluns in different order.

It is not clear to me what you mean with the linear bezier case. Is this case? You can find some results on Bézier curves on SearchOnMath that can helps you.

Last edited:

## What is a Linear Bezier curve projection?

A Linear Bezier curve projection is a mathematical technique used to approximate a smooth curve by connecting a series of straight line segments. It is commonly used in computer graphics and animation to create smooth, curved shapes.

## How is a Linear Bezier curve projection calculated?

A Linear Bezier curve projection is calculated using a set of control points, which are used to define the shape of the curve. The curve is then approximated by connecting these control points with straight line segments.

## What are the advantages of using a Linear Bezier curve projection?

One of the main advantages of using a Linear Bezier curve projection is that it allows for smooth, curved shapes to be created using only straight line segments. This makes it a more efficient and simpler method compared to other curve approximation techniques.

## What are the limitations of a Linear Bezier curve projection?

One limitation of a Linear Bezier curve projection is that it can only approximate curves, and may not be able to accurately represent more complex shapes. Additionally, the number of control points used can greatly affect the accuracy of the curve.

## How is a Linear Bezier curve projection used in computer graphics?

A Linear Bezier curve projection is commonly used in computer graphics to create smooth, curved shapes such as circles, arcs, and other complex shapes. It is also used in animation to create fluid and natural movements of objects.

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