A Bézier curve is a parametric curve frequently used in computer graphics and related fields...

A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.

In vector graphics, Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs,[note 1] are combinations of linked Bézier curves. Paths are not bound by the limits of rasterized images and are intuitive to modify.

Bézier curves are also used in the time domain, particularly in animation, user interface[note 2] design and smoothing cursor trajectory in eye gaze controlled interfaces.[1] For example, a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators or interface designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.

This also applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear.

The mathematical basis for Bézier curves — the Bernstein polynomial — has been known since 1912, but its applicability to graphics was not realized for another half century. Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies at Renault. The study of these curves was however first developed in 1959 by mathematician Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bézier curves at Citroën, another French automaker.[2]

[Note 1]: Image manipulation programs such as Inkscape, Adobe Photoshop, and GIMP.

[Note 2]: In animation applications such as Adobe Flash, Adobe After Effects, Microsoft Expression Blend, Blender, Autodesk Maya and Autodesk 3ds max.

[1]: Biswas, Pradipta; Langdon, Pat (2015-04-03). "Multimodal Intelligent Eye-Gaze Tracking System". International Journal of Human-Computer Interaction 31 (4): 277–294. doi:10.1080/10447318.2014.1001301. ISSN 1044-7318.

[2]: Gerald E. Farin; Josef Hoschek; Myung-Soo Kim (2002). Handbook of Computer Aided Geometric Design. Elsevier. pp. 4–6. ISBN 978-0-444-51104-1.

Dude, I love Bézier curves!

I can't explain why… but threads like this make me incredibly hard, like diamond.

I have a question about this picture.
Does the girl hear with both sets of ears?

Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the convex hull of its control points, the 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.[citation needed]

Quadratic and cubic Bézier curves are most common. Higher degree curves are more computationally expensive to evaluate. When more complex shapes are needed, low order Bézier curves are patched together, producing a composite Bézier curve. A composite Bézier curve is commonly referred to as a "path" in vector graphics languages (like PostScript), vector graphics standards (like SVG) and vector graphics programs (like Adobe Illustrator, CorelDraw and Inkscape). To guarantee smoothness, the control point at which two curves meet must be on the line between the two control points on either side.[citation needed]
Sorry for lack of citations, just trust me, I'm an expert.

Picture: Cubic Bézier curve with four control points


I really wouldn't know.

The simplest method for scan converting (rasterizing) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments. 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. A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line segment to within a small tolerance. If not, the curve is subdivided parametrically into two segments, 0 ≤ t ≤ 0.5 and 0.5 ≤ t ≤ 1, and the same procedure is applied recursively to each half. There are also forward differencing methods, but great care must be taken to analyse error propagation.[citation needed]

Analytical methods where a Bézier is intersected with each scan line involve finding roots of cubic polynomials (for cubic Béziers) and dealing with multiple roots, so they are not often used in practice.[citation needed]

Picture: The basis functions on the range t in [0,1] for cubic Bézier curves: blue: y0 = (1 − t)3, green: y1 = 3(1 − t)2 t, red: y2 = 3(1 − t) t2, and cyan: y3 = t3.

According to the "lore", elin use both pairs of ears to hear different things - they use their human ears to understand the language of humans and other intelligent beings, while their animal ears help them hear the voice of nature.

The exact formulas didn't translate here very well, so let's jump straight to NURBS until I figure that out.

Non-uniform rational Basis spline (NURBS) is a mathematical model commonly used in computer graphics for generating and representing curves and surfaces. It offers great flexibility and precision for handling both analytic (surfaces defined by common mathematical formulae) and modeled shapes. NURBS are commonly used in computer-aided design (CAD), manufacturing (CAM), and engineering (CAE) and are part of numerous industry wide standards, such as IGES, STEP, ACIS, and PHIGS. NURBS tools are also found in various 3D modelling and animation software packages.

Picture: A NURBS curve.


Luckily we have an expert here in the lecture hall.

When does the next lecture start?

I'm ready!

They can be efficiently handled by the computer programs and yet allow for easy human interaction. NURBS surfaces are functions of two parameters mapping to a surface in three-dimensional space. The shape of the surface is determined by control points. NURBS surfaces can represent, in a compact form, simple geometrical shapes. T-splines and subdivision surfaces are more suitable for complex organic shapes because they reduce the number of control points twofold in comparison with the NURBS surfaces.

In general, editing NURBS curves and surfaces is highly intuitive and predictable. Control points are always either connected directly to the curve/surface, or act as if they were connected by a rubber band. Depending on the type of user interface, editing can be realized via an element’s control points, which are most obvious and common for Bézier curves, or via higher level tools such as spline modeling or hierarchical editing.

Animation: Three-dimensional NURBS surfaces can have complex, organic shapes. Control points influence the directions the surface takes. The outermost square below delineates the X/Y extents of the surface.

Can you teach us how NURBS and Bézier curves are actually calculated?

...

I don't know how to LaTex and whether it's even enabled. I can post the equations as pictures, but explaining them in text is cumbersome.

Here's the explicit definition and as noted earlier it includes Bernstein polynomial, which may wish to look more into.

… explicit definition of a Bezier curve, that is.

A NURBS curve is defined by its order, a set of weighted control points, and a knot vector.[4] NURBS curves and surfaces are generalizations of both B-splines and Bézier curves and surfaces, the primary difference being the weighting of the control points, which makes NURBS curves rational (non-rational B-splines are a special case of rational B-splines). By using a two-dimensional grid of control points, NURBS surfaces including planar patches and sections of spheres can be created. These are parametrized with two variables (typically called s and t or u and v).

[4]: Bio-Inspired Self-Organizing Robotic Systems. p. 9. Retrieved 2014-01-06.

Mah nigga.
Elins are the best.

...

i want to use my dick to explore an elin honeyhole

No lessons for more than 24 hours—What happened?