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- De Casteljau's Algorithm and Bézier Curves
- Finding a Point on a Bézier Curve: De Casteljau's Algorithm
- Bézier curve
- Bezier Curves Made Simple
- Cubic Bezier Curve Implementation in C

## De Casteljau's Algorithm and Bézier Curves

Bezier curves are the most fundamental curves, used generally in computer graphics and image processing. These curves are mainly used in interpolation, approximation, curve fitting, and object representation. In this article, I will demonstrate, in a very simple and straightforward way, how one can construct these curves and make use of them. Bezier curves are parametric curves which are pretty much customizable and smooth. They are well suited for many applications. People say that at the same time the same development took place during the research of Ford. There is still a confusion about who found it first. Because of my imaging background, my article will mainly focus on interpolation and curve fitting. In interpolation, what one would simply like to do is to find unknown points using known values. This way, a discrete case can be represented with a more continuous structure, and we can have a well defined curve for missing points. The curve is initialized with certain data points, and it tries to generate new ones that are approximating or interpolating the old values. The idea is demonstrated below:. The objective here is to find points in the middle of two nearby points and iterate this until we have no more iterations. The new values of points will give us the curve. The famous Bezier equation is the exact formulation of this idea. Here is the algorithm:. This value remains constant for the rest of the steps. Now, I will give formulas for common, special cases that can be helpful in certain applications. The code of the article does not demonstrate any of them, but it uses the generalized formula. So, let me start with the generalized formula:. For the sake of simplicity and convention used in this article and code, it is better to represent this formula as:. What this equation tells us is nothing but the formulation of the above algorithm the mid-point iterations.## Finding a Point on a Bézier Curve: De Casteljau's Algorithm

Help to translate the content of this tutorial to your language! Bezier curves are used in computer graphics to draw shapes, for CSS animation and in many other places. They are a very simple thing, worth to study once and then feel comfortable in the world of vector graphics and advanced animations. A bezier curve is defined by control points. Points are not always on curve. The curve order equals the number of points minus one. A curve is always inside the convex hull of control points:. If convex hulls do not intersect, then curves do not either. Checking the intersection or convex hulls is much easier, because they are rectangles, triangles and so on see the picture abovemuch simpler figures than the curve. The main value of Bezier curves for drawing — by moving the points the curve is changing in intuitively obvious way. After some practice it becomes obvious how to place points to get the needed curve. And by connecting several curves we can get practically anything. Draw control points. In the demo above they are labeled: 123. In the demo above they are brown. The parameter t moves from 0 to 1. In the example above the step 0. On each brown segment we take a point located on the distance proportional to t from its beginning. As there are two segments, we have two points. Connect the points. On the picture below the connecting segment is painted blue. Now in the blue segment take a point on the distance proportional to the same value of t. On pictures above that point is red. As t runs from 0 to 1every value of t adds a point to the curve. The set of such points forms the Bezier curve. As the algorithm is recursive, we can build Bezier curves of any order, that is: using 5, 6 or more control points. But in practice many points are less useful. Usually we take points, and for complex lines glue several curves together. To specify a Bezier curve, control points are used. As we can see, they are not on the curve, except the first and the last ones.

## Bézier curve

Paths are not bound by the limits of rasterized images and are intuitive to modify. This also applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear. The first and last control points are always the end points of the curve; however, the intermediate control points if any generally do not lie on the curve. The sums in the following sections are to be understood as affine combinationsthe coefficients sum to 1. The curve is given by. Rearranging the preceding equation yields:. As t increases from 0 to 1, the curve departs from P 0 in the direction of P 1then bends to arrive at P 2 from the direction of P 1. The curve starts at P 0 going toward P 1 and arrives at P 3 coming from the direction of P 2. Usually, it will not pass through P 1 or P 2 ; these points are only there to provide directional information. The distance between P 1 and P 2 determines "how far" and "how fast" the curve moves towards P 1 before turning towards P 2. For some choices of P 1 and P 2 the curve may intersect itself, or contain a cusp. Then to start. Application of the binomial theorem to the definition of the curve followed by some rearrangement will yield:. Note that the empty product is 1. As t varies from 0 to 1, B t describes a straight line from P 0 to P 1. For higher-order curves one needs correspondingly more intermediate points. Therefore, new control points are [16]. The concept of degree elevation can be repeated on a control polygon R to get a sequence of control polygons RR 1R 2and so on. After r degree elevations, the polygon R r has the vertices P 0,rP 1,rP 2,rThe expression can be extended by using number systems besides reals for the weights. For curves with points and weights on a circle, the weights can be scaled without changing the curve's shape. As the curve is completely contained in the convex hull of its control pointsthe points can be graphically displayed and used to manipulate the curve intuitively. Affine transformations such as translation and rotation can be applied on the curve by applying the respective transform on the control points of the curve. Higher degree curves are more computationally expensive to evaluate. This guarantees that the resultant curve is C1 continuous, meaning that the first derivative is continuous everywhere. However, this does not guarantee that the rasterized output looks sufficiently smooth, because the points may be spaced too far apart. Conversely it may generate too many points in areas where the curve is close to linear.

## Bezier Curves Made Simple

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. Algorithms like Bresenham's work pixel-by-pixel to rasterize a line, and can be optimized see Po-Han Lin's solution. A variation of Bresenham's Algorithm works with quadratic functions like circles, ellipses, and parabolas, so it should work with quadratic Bezier curves too. Here's the code from Alois Zingl's site for plotting any quadratic Bezier curve. The first routine subdivides the curve at horizontal and vertical gradient changes:. As Mike 'Pomax' Kamermans has noted, the solution for cubic Bezier curves on the site is not complete; in particular, there are issues with antialiasing cubic Bezier curves, and the discussion of rational cubic Bezier curves is incomplete. You can use De Casteljau's algorithm to subdivide a curve into enough pieces that each subsection is a pixel. To make practical use of this equation, you can input about T values between 0. This results in a set of points guaranteed to be along the Quadratic Curve. Calculating points along the curve is probably over-sampling some calculated points will be at the same pixel coordinate so you will want to de-duplicate the points until the set represents unique pixel coordinates along the curve. The thing to realise here is that "line segments", when created small enough, are equivalent to pixels. Bezier curves are not linearly traversible curves, so we can't easily "skip ahead to the next pixel" in a single step, like we can for lines or circular arcs. You could, of course, take the tangent at any point for a t you already have, and then guess which next value t' will lie a pixel further. However, what typically happens is that you guess, and guess wrong because the curve does not behave linearly, then you check to see how "off" your guess was, correct your guess, and then check again. Repeat until you've converged on the next pixel: this is far, far slower than just flattening the curve to a high number of line segments instead, which is a fast operation. If you pick the number of segments such that they're appropriate to the curve's length, given the display it's rendered to, no one will be able to tell you flattened the curve. There are ways to reparameterize Bezier curves, but they're expensive, and different canonical curves require different reparameterization, so that's really not faster either. What tends to be the most useful for discrete displays is to build a LUT lookup table for your curve, with a length that works for the size the curve is on the display, and then using that LUT as your base data for drawing, intersection detection, etc. First of all, I'd like to say that the fastest and the most reliable way to render bezier curves is to approximate them by polyline via adaptive subdivision, then render the polyline. Approach by markE with drawing many points sampled on the curve is rather fast, but it can skip pixels. Here I describe another approach, which is closest to line rasterization though it is slow and hard to implement robustly. This algorithm works until time parameter exceeds one. Also note that it has severe issues with curves exactly touching a side of a pixel. I suppose it is solvable with a special check. Full code is available here. It uses loadbmp. How are we doing? Please help us improve Stack Overflow. Take our short survey. Learn more. Asked 4 years, 8 months ago. Active 9 months ago. Viewed 6k times. It's a very good question, but I'm not sure it has an answer. A Bezier isn't calculated from its X or Y coordinate, but from an independent variable T that isn't correlated with either.

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