1) writes control information into control port 278. Such information comprises three control bits: one is a reset bit; the second is a command bit (the 17th bit line of the data bus which differentiates between commands and parameters); and the third is used by CPU 800 to turn on LED DS2. I keep on getting e-mails on my Bezier code and in between them, how to draw the Bezier Handles. It is very simple and it is pure trigonometry knowledge to pull it off. To draw the handle is very easy. Tap or Click where you would like the first anchor point is going to be, then drag towards the desired curvature is achieved.
In order to draw curvy surface we implement Bezier curve algorithm. In drawing Bezier we first define n+1 control point pk = (xk, yk, zk) with k varying from 0 to n. And the coordinate points are blended to produce following position vectors that define the path of an approximation Bezier polynomial function between p0 and pn.
The Bezier blending functions BEZk,n(u) are the Bernstein polynomials: BEZk,n(u)=C(n,k)uk(1-u)n-k Where the C(n, k) are binomial coefficients. Equivalently, we can define Bezier blending functions with the recursion calculation. BEZk,n (u) = (1-u) BEZk,n-1(u)+u BEZk-1,n-1(u), n>k≥1 With BEZk,k = uk , and BEZ0,k = (1-u)k.
C Source Code
Note: to run this code in your machine with Code::blocks IDE, add a link
library libgdi32.a (it is usually inside MinGWlib ) in linker setting.
Output
SHARE Bezier Curves and Bezier Surfaces generation with C/C++ in Code::BlocksWritten by Paul Bourke December 1996
Contribution by Prashanth Udupa on Bezier Surfaces in VTK Designer 2:Bezier_VTKD2.pdf
The Bézier surface is formed as the Cartesian product ofthe blending functions of two orthogonal Bézier curves.
Where Pi,j is the i,jth control point. There are Ni+1 and Nj+1 controlpoints in the i and j directions respectively.
The corresponding properties of the Bézier curve apply to the Bézier surface. - The surface does not in general pass through the control points except for the corners of the control point grid. - The surface is contained within the convex hull of the control points.
Along the edges of the grid patch the Bézier surface matches that ofa Bézier curve through the control points along that edge.
Closed surfaces can be formed by setting the last controlpoint equal to the first. If the tangents also match between the firsttwo and last two control points then the closed surface will have first order continuity.
While a cylinder/cone can be formed from a Bézier surface, it isnot possible to form a sphere.
C Source Example
The following source code generates the surface shown inthe first example above. It is provided for illustrationonly, the headers and prototype files are not given.
Bézier Blending Function
Beziercode 1 264
This function computes the blending function as used in theBézier surface code above. It is written for clarity, notefficiency. Normally, if the number of control points isconstant, the blending function would be calculated once foreach desired value of mu.
Written by Paul Bourke Original: April 1989, Updated: December 1996
The following describes the mathematics for the so called Bézier curve. It isattributed and named after a French engineer, Pierre Bézier,who used them for the body design of the Renault car in the 1970's.They have since obtained dominance in the typesetting industry andin particular with the Adobe Postscript and font products.
Consider N+1 control points pk (k=0 to N) in 3 space. The Bézier parametriccurve function is of the form Aiseesoft mac video converter ultimate 9 2 62 bit.
B(u) is a continuous function in 3 space defining the curve with N discretecontrol points Pk. u=0 at the first control point (k=0) and u=1 at the lastcontrol point (k=N).
Notes:
The curve in general does not pass through any of the control points except the first and last. From the formula B(0) = P0 and B(1) = PN.
The curve is always contained within the convex hull of the control points, it never oscillates wildly away from the control points.
If there is only one control point P0, ie: N=0 then B(u) = P0 for all u.
If there are only two control points P0 and P1, ie: N=1 then the formula reduces to a line segment between the two control points.
the term
is called a blending function since it blends the control points to form the Bézier curve.
The blending function is always a polynomial one degree less than the number of control points. Thus 3 control points results in a parabola, 4 control points a cubic curve etc.
Closed curves can be generated by making the last control point the same as the first control point. First order continuity can be achieved by ensuring the tangent between the first two points and the last two points are the same.
Adding multiple control points at a single position in space will add more weight to that point 'pulling' the Bézier curve towards it.
As the number of control points increases it is necessary to have higherorder polynomials and possibly higher factorials. It is common therefore topiece together small sections of Bézier curves to form a longer curve. Thisalso helps control local conditions, normally changing the position of onecontrol point will affect the whole curve. Of course since the curve starts andends at the first and last control point it is easy to physically match thesections. It is also possible to match the first derivative since the tangentat the ends is along the line between the two points at the end. Second order continuity is generally not possible.
Except for the redundant cases of 2 control points (straight line), it is generally not possible to derive a Bézier curve that is parallel to another Bézier curve.
A circle cannot be exactly represented with a Bézier curve.
It isn't possible to create a Bézier curve that is parallel to another, except in the trivial cases of coincident parallel curves or straight line Bézier curves.
Special case, 3 control points
B(u) = P0 * ( 1 - u ) 2 + P1 * 2 * u ( 1 - u ) + P2 u2
Special case, 4 control points
B(u) = P0 * ( 1 - u )3 + P1 * 3 * u * ( 1 - u )2 + P2 * 3 * u2 * ( 1 - u ) + P3 * u3
Bézier curves have wide applications because they are easy to compute and verystable. There are similar formulations which are also called Bézier curveswhich behave differently, in particular it is possible to create a similarcurve except that it passes through the control points. See also Spline curves. Examples
The pink lines show the control point polygon, the grey lines theBézier curve.
The degree of the curve is one less than the number of controlpoints, so it is a quadratic for 3 control points.It will always be symmetric for a symmetric control point arrangement.
The curve always passes through the end points and is tangent tothe line between the last two and first two control points.This permits ready piecing of multiple Bézier curves togetherwith first order continuity.
The curve always lies within the convex hull of the control points.Thus the curve is always 'well behaved' and does not oscillating erratically.
Closed curves are generated by specifying the first point the same as the last point. If the tangents at the first and last points matchthen the curve will be closed with first order continuity.In addition, the curve may be pulled towards a control point byspecifying it multiple times.
C source
Written by Paul Bourke March 2000
Given four points p0, p1, p2, and p3in 3D space the cubic Bézier curve is defined as
p(t) = a t3 + b t2 + c t + p0
where t ranges from 0 (the start of the curve, p0) to 1 (the end of the curve, p3). The vectors a, b, c are given as follows:
Beziercode 1 26th
c = 3 (p1 - p0)
b = 3 (p2 - p1) - c
a = p3 - p0 - c - b
In the following examples the green markers correspond to p0and p3 of each section. The blue markers correspond top1 and p2. The grey curve is theBézier curve sampled 20 times, the samples are shown in red.The coordinates for each vertex is shown on the right.
Example 1 This is a single minimum piece of a piecewise Bézier curve.It is defined by 4 points, the curve passes through the two end points.The tangent at the end points is along the line to the middle two points.
0 0 1 0.5 0 1 1 0 0.5 1 0 0
Example 2 Multiple curve pieces can be joined together to form longer continuouscurves. The curve is made continuous by the setting the tangents thesame at the join. Note that each piece of the curve is defined by tranging from 0 to 1.
0 0 1 0.5 0 1 1 0 0.5 1 0 0
1 0 0 1 0 -0.5 2 0 0 2 0 0.5
Beziercode 1 268
Example 3 By changing the tangent points between two curve pieces, sharp transitionscan be created.
0 0 1 0.5 0 1 1 0 0.5 1 0 0
1 0 0 1.5 0 0 2 0 0 2 0 0.5
Example 4 The 'strength' at the end points is controlled by the length of thetangent lines. The longer the line the more effect that tangent has.If the curve is being used for animation steps then the strength alsocontrols the velocity, note the samples shown in red are further apartfor the long tangent vectors.
0 0 1 1.75 0 1 1 0 0.5 1 0 0
1 0 0 1 0 -0.5 2 0 -0.5 2 0 1
Example 5 Straight line geometry can readily be made by aligning the tangentvectors along the line. While this may seem a frivolous use, it canbe put to good effect in animations applications. By adjusting thetangent points p1 and p2 thevelocity along the line can be controlled.
0 0 1 0.25 0 1 0.75 0 1 1 0 1
1 0 1 1 0 0.75 1 0 0.25 1 0 0
Notes
Piecewise cubic Bézier curves like their most general Bézier counterparts cannot exactly represent a circle.
Except in the trivial case of a straight line, it isn't possible to create a Bézier curve that is parallel to another.
There is no closed solution to finding the closest point on a Bézier curve to another point. The usual method is some kind of subdivision of t until some error tolerance is met.
0to p5, in order to use the Piecewise Cubic Bézierfor each section (between points pi and pi+1)one needs to find the tangent vectors shown in red.Note that for continuity between the points the tangent vector atthe end of one piece is the negative of the tangent at thestart of the next piece.
Beziercode 1 263
In this first case the tangent vectors are just the differences betweensubsequent keyframe points. So for example, for the segment between p1 and p2 the four points use for the Bézierwould be p1, p2, 2p2-p3, p2. Depending on the lengthscaling for the tangent vectors, the resulting Bézier curvebetween points p1 and p3 is shown in blue.
A generally better method is shown below, again one needs to find thered tangent vectors. The exact implementation will be left up to thereader but the approach I've used is to find the cross product betweenthe vectors to each neighbour, that is a vector coming outof the page (or into the page) in the diagram below. The tangentvectors (red) are found by taking the cross product of that withthe green normal vectors. The main reason for using this approachis that it overcomes a mirror symmetry problem that occurs if onesimple tries to rotate the green normal vectors +- 90 degrees.Note that the case of 3 collinear points needs to be treated as a specialcase.
An improvement by Lars Jensen is illustrated below. It uses a normalthat bisects the two vectors to the neighbouring points along with wayof limiting the tangent lengths.
The only remaining comment is how one deals with the first and lastpoint, normally there are some ad hoc approaches that are applicationspecific.
