Exploration of the Bézier curve

Special case of the Bézier curve is as follows.

As pre requisite recommend reading about the Bézier curve and De_Casteljau's algorithm which provides geometric interpretation to curve construction here.

Where a curvature in a local coordinate system has no inflection points and can be described by one local maximum where this is defined in terms of a local y position where a relative coordinate system defined by the plane between points P₀ and P₃), and given two points P₁ and P₂ such that the relative positions y_1r and y_2r on such plane are equidistant from the tangent of the line P₀ and P₃. A local y_r maximum is given by

where and θ_P0,P3 represents the angle of the tangent between the segment (P₀,P₃)

Proof:

We can rotate and translate the given coordinate axis such that such that the P₁ represents a given origin 0,0 on a relative coordinate system, and furthermore we can ensure that rotation of the coordinate plane also matches the tangent line of line P₀, P₃, so that y₀, and y₃ are zero while x₀ and x₃ are not equal. By assumptions above

y₁ and y₂ are equal and remain so under coordinate changes so that are Bézier equation becomes for the y relative in the parametric form  

where y_r(t) is the parametric form of the Bézier equation in our rotated and translated coordinate plane. Keep in mind that the parametric form on the rotated plane preserve the magnitude and the relative positions of P₀, P₁, P₂, and P₃.

taking the derivative with respect to t we have

as long as x_r(t) is not simultaneously 0 at t, the derivative dy_r/ dx_r takes the form dy_r / dt / dx_r / dt

setting this equal to zero means that we are concerned with the zero in the numerator,

so we solve for y_r '(t) = 0

In this case a maximum slope on the relative coordinate system is found at

t = 1/2

Substituting t = 1/2 into the equation y_r (t) means

or

we need rotate our coordinate P₁ to relative coordinate along side providing a translation of such coordinates as this provides a sense of scale in the rotated coordinate system on the relative coordinate system relative to original coordinate positions, or this is to say, expressing the local y in terms of control point P₁.

A translation can be defined with an affine transformation.

In order to express P₀ as a point of origin on our given transformed coordinate system.

or in the case of our coordinate P₀ this is written  

while a rotation in coordinates is defined by

  

Thus we can see that P₀ is zero under the series of transforms with

  

where this resultant vector (0,0,1) time the rotation matrix is also zero.

We can see that P₃ is zero on the transformed y relative under the series of transforms with the affine translation.

rt*tr*P₃  

for the affine transform

and for rotation

Here examining the y relative component of the transformed P₃ position we should notice that y relative coordinate component is

geometrically one can see this from the following graph showing the rotation axis and original axis

The graph will demonstrates points are already affine translated

Here where θ in the graph equals the given rotation angle defined above we can see the y' components cancel when subtracted which results from the rotation matrix transform and the point P₃ - P₀ lay on the zero axis of the new rotated coordinate system.

Similarly under transformation y_1r can be expressed in terms of P₁ as

Thus for any coordinate point we have a transform which is

  

  

substituting eq. (2) into (1) we have

  

We can check that the t = 1/2 doesn't simultaneously equal to zero here, for x_r'(t).

If we wanted to find the original positions x(t), y(t) for t = 1/2 we can either re transform coordinates

using the inverse of the transform matrix above, or we could compute the Bézier curve for t = 1/2

which should remain relatively invariant in so far as the arrangement of points overall for the given

curvature. In other words, rotation and translation only changes the position of all points equally not the relative arrangement of points for such curvature. Why is this?

The Bézier curve geometrically computes on the basis of the relative arrangement of points in determining the shape of the curve, the relative arrangement of control points control the shape of the curvature, but this remains invariant (unchanged) under rotation and translation.

In other words, if the relative angles remain the same between control points alongside preservation of magnitude, these will determine the points on the curve which should be relatively the same irrespective of choice in coordinate plane. Geometrically speaking the points of the polygon formed initially by the control points are the same relatively speaking in so far as distance magnitudes, and the t parameterization on the relative coordinate systems trace the same relative distance when constructing a solution on the basis of relative spatial subdivisions. In the case of the cubic, a triangle is formed irrespective axis at point t₀ on any axis consisting of the same side lengths, and this in turn leads to a line subdivision of the same magnitude line in length irrespective of axis at t₀ which determines the point on the Bézier curve. The points themselves are variant under rotation and translation but not the shape of the curve.

It would be worth noting that in the original coordinate system the relative maximum computed t = 1/2 may not the maximum of the curve, or a given slope = 0 at such point, but the relative coordinate