Question

A cubic Bézier curve is a parametric curve in \(\mathbb{R}^{2}\) of the form $$ A \cdot(i+1-u)^{3}+B \cdot 3(i+1-u)^{2}(u-i)+C \cdot 3(i+1-u)(u-i)^{2}+D \cdot(u-i)^{3}, $$ where \(A, B, C, D\) are points in \(\mathbb{R}^{2}, i \in \mathbb{N}\), and the parameter \(u\) runs through the real interval \([i, i+1]\).

Short answer

The Bézier curve smoothly transitions from point \(A\) to \(D\) as \(u\) varies from \(i\) to \(i+1\).

Step-by-step solution

Understand the Bézier Curve Equation

The given cubic Bézier curve is expressed as a parametric equation in terms of points \(A, B, C, D\) which are control points in \(\mathbb{R}^2\). The parameter \(u\) varies between \(i\) and \(i+1\), which suggests the curve segment corresponds to this interval. The equation is a weighted blend of the control points, influenced by the parameters \((i+1-u)^3\), \(3(i+1-u)^2(u-i)\), \(3(i+1-u)(u-i)^2\), and \((u-i)^3\).

Recognizing the Parametric Intervals

Notice that the parameter \(u\) is defined in the interval \([i, i+1]\). This means for each integer \(i\), the parameter \(u\) runs from \(i\) to \(i+1\). Throughout this interval, the Bézier curve segment transitions from the influence of point \(A\) towards point \(D\), with \(B\) and \(C\) controlling the transition.

Understanding Coefficients of Control Points

The coefficients \((i+1-u)^3\), \(3(i+1-u)^2(u-i)\), \(3(i+1-u)(u-i)^2\), and \((u-i)^3\) are derived from the binomial expansion used in Bézier curves and they are Bernstein polynomials. These polynomials control how much each control point influences the final position of the curve as \(u\) changes from \(i\) to \(i+1\).

Analyze Edge Cases of the Interval

When \(u = i\), the equation simplifies to \(A\) as the dominant term due to the term influenced by \((i+1-u)^3\). Similarly, at \(u = i+1\), the equation simplifies to \(D\), dominated by the term \((u-i)^3\). Therefore, the curve starts at point \(A\) and ends at point \(D\) for the interval \([i, i+1]\).

Consider Intermediate Values of u

For values of \(u\) between \(i\) and \(i+1\), the curve smoothly interpolates between points \(A\) and \(D\) while points \(B\) and \(C\) shape the curvature. The particular form of the curve is controlled by how these intermediate points adjust the influence of each control point over the interval.

Key concepts

Parametric Equation

In the world of mathematics and computer graphics, a parametric equation allows us to express geometric shapes in terms of a parameter. Rather than describing a path or surface directly in terms of axes, we use an additional parameter to represent the curve.
Consider a cubic Bézier curve. It is expressed as a parametric equation involving a parameter, often denoted as \( u \). The equation pairs this parameter with control points to define the curve. As \( u \) changes, the position on the curve changes.

Unlike regular equations, parametric equations involve one or more independent parameters. They simplify the description of complex shapes. Instead of decomposing into individual coordinates, we refer to a complete position in the space with just one variable.

• Parametric equations are particularly useful for animation and graphics.
• The parameter \( u \) typically ranges over a specified interval, defining the part of the curve we observe.

For Bézier curves, the value of \( u \) goes from starting point \( i \) to ending point \( i+1 \), creating segments of the curve.

Control Points

Control points are fundamental building blocks in Bézier curves. They are not actual points on the curve but rather influence its shape and direction. Each Bézier curve is defined by a specific number of control points. For a cubic Bézier, we use four: \( A, B, C, \) and \( D \).
These points determine how the curve bends and twists between its start and end.

In a cubic Bézier curve, control points play different roles:

• Start and End Points: They anchor the curve at \( A \) and \( D \), marking where it begins and ends.
• Intermediate Control Points: \( B \) and \( C \) influence the path the curve takes through its midpoint.

The positions of these control points directly affect the curve's trajectory.
By adjusting them, we can refine the curve for different applications, creating smooth arcs, sharp turns, or gentle waves. Understanding control points gives us the power to tailor the curve's shape precisely.

Bernstein Polynomials

Bernstein polynomials serve as the coefficients in the Bézier curve's equation. They provide the mechanism by which control points are blended into the curve as the parameter \( u \) changes. These polynomials are defined in terms of the binomial coefficients and the parameter itself.
In a cubic Bézier curve, the Bernstein polynomials are:

• \((i+1-u)^3\)
• \(3(i+1-u)^2(u-i)\)
• \(3(i+1-u)(u-i)^2\)
• \((u-i)^3\)

Each of these polynomials acts as a weighting factor for its corresponding control point.

They determine the degree of influence each point asserts over a given position on the curve:

• The higher the polynomial's value, the more control that point has over the curve.
• Summing the weights always results in 1, ensuring the curve remains a valid path between points \( A \) and \( D \).

This property of Bernstein polynomials empowers us to smoothly dissolve the transition of the curve from the influence of the starting control point \( A \) to the ending point \( D \).