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# 5 Bezier curves

Each interior node of a cubic spline may be given a
direction prefix or suffix `{dir}`

: the direction of the pair
`dir`

specifies the direction of the incoming or outgoing tangent,
respectively, to the curve at that node. Exterior nodes may be
given direction specifiers only on their interior side.

A cubic spline between the node *z_0*, with postcontrol point
*c_0*, and the node *z_1*, with precontrol point *c_1*,
is computed as the Bezier curve

As illustrated in the diagram below, the third-order midpoint (*m_5*)
constructed from two endpoints *z_0* and *z_1* and two control points
*c_0* and *c_1*, is the point corresponding to *t=1/2* on
the Bezier curve formed by the quadruple (*z_0*, *c_0*,
*c_1*, *z_1*). This allows one to recursively construct the
desired curve, by using the newly extracted third-order midpoint as an
endpoint and the respective second- and first-order midpoints as control
points:

Here *m_0*, *m_1* and *m_2* are the first-order
midpoints, *m_3* and *m_4* are the second-order midpoints, and
*m_5* is the third-order midpoint.
The curve is then constructed by recursively applying the algorithm to
(*z_0*, *m_0*, *m_3*, *m_5*) and
(*m_5*, *m_4*, *m_2*, *z_1*).

In fact, an analogous property holds for points located at any
fraction *t* in *[0,1]* of each segment, not just for
midpoints (*t=1/2*).

The Bezier curve constructed in this manner has the following properties:

- It is entirely contained in the convex hull of the given four points.
- It starts heading from the first endpoint to the first control point and finishes heading from the second control point to the second endpoint.

The user can specify explicit control points between two nodes like this:

draw((0,0)..controls (0,100) and (100,100)..(100,0));

However, it is usually more convenient to just use the
`..`

operator, which tells `Asymptote`

to choose its own
control points using the algorithms described in Donald Knuth’s
monograph, The MetaFontbook, Chapter 14.
The user can still customize the guide (or path) by specifying
direction, tension, and curl values.

The higher the tension, the straighter the curve is, and the more it approximates a straight line. One can change the spline tension from its default value of 1 to any real value greater than or equal to 0.75 (cf. John D. Hobby, Discrete and Computational Geometry 1, 1986):

draw((100,0)..tension 2 ..(100,100)..(0,100)); draw((100,0)..tension 3 and 2 ..(100,100)..(0,100)); draw((100,0)..tension atleast 2 ..(100,100)..(0,100));

In these examples there is a space between `2`

and `..`

.
This is needed as `2.`

is interpreted as a numerical constant.

The curl parameter specifies the curvature at the endpoints of a path (0 means straight; the default value of 1 means approximately circular):

draw((100,0){curl 0}..(100,100)..{curl 0}(0,100));

The `MetaPost ...`

path connector, which requests, when possible, an
inflection-free curve confined to a triangle defined by the
endpoints and directions, is implemented in `Asymptote`

as the
convenient abbreviation `::`

for `..tension atleast 1 ..`

(the ellipsis `...`

is used in `Asymptote`

to indicate a
variable number of arguments; see section Rest arguments). For example,
compare

draw((0,0){up}..(100,25){right}..(200,0){down});

with

draw((0,0){up}::(100,25){right}::(200,0){down});

The `---`

connector is an abbreviation for ```
..tension atleast
infinity..
```

and the `&`

connector concatenates two paths, after
first stripping off the last node of the first path (which normally
should coincide with the first node of the second path).

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