blocks|key|1552647|text|更清楚的是，ID2D1GeometrySink，不支持样条，而支持三次bezier曲线，这些曲线可以组合成样条。相反，你可以从你的样条数据中得到b-曲线，然后把它们添加到你的几何中。|type|unstyled|depth|inlineStyleRanges|offset|length|style|BOLD|entityRanges|data|1552648|这张图片简单地解释了该算法：|1552649|📷|atomic|1552650|。|1552651|一篇简短和很好的解释的文章可以找到这里。你可以分裂你的样条，直到控制点重叠，程度可以降低，甚至直到所有的曲线都是平到成线为止。最后一件事并不是个坏主意，因为你的硬件不知道曲线，所以你通过的曲线会被转换成线条。当+you+执行此转换时，您可以确定平面度的公差，并避免难看的边缘。|1552652|entityMap|0|IMAGE|mutability|IMMUTABLE|imageUrl|https://developer.qcloudimg.com/http-save/yehe-900000/f5c9d30e2ed68abfe67a4f0437b3a3dc.gif|imageAlt|1|LINK|MUTABLE|url|http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node18.html^0|6|H|0|0|0|1|0|0|0|2X|5|H|2|1|0^^$0|@$1|2|3|4|5|6|7|12|8|@$9|13|A|14|B|C]]|D|@]|E|$]]|$1|F|3|G|5|6|7|15|8|@]|D|@]|E|$]]|$1|H|3|I|5|J|7|16|8|@]|D|@$9|17|A|18|1|19]]|E|$]]|$1|K|3|L|5|6|7|1A|8|@]|D|@]|E|$]]|$1|M|3|N|5|6|7|1B|8|@$9|1C|A|1D|B|C]]|D|@$9|1E|A|1F|1|1G]]|E|$]]|$1|O|3|-4|5|6|7|1H|8|@]|D|@]|E|$]]]|P|$Q|$5|R|S|T|E|$U|V|W|-4]]|X|$5|Y|S|Z|E|$10|11]]]]

Direct2D, more clear ID2D1GeometrySink, doesn't support splines, but cubic bezier curves which can be put together to a spline. In the opposite, you can gain b-curves out of your spline data and just add them to your geometry.

The algorithm is simply explained by this picture: <img src="https://i.stack.imgur.com/I1s3N.gif" alt="curve substitution">.

An article for short and well explanation can be found <a href="http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node18.html" rel="nofollow noreferrer">here</a>.
You can split your spline until control points overlap and the degree can be lowered, even until all curves are flat enough to be lines. Last thing isn't a bad idea because your hardware doesn't know curves, so your passed curves get converted to lines later anyway. When you do this conversion, you can determine the tolerance for flatness and avoid ugly edges.

blocks|key|397063|text|除非您已经有基本NURBS操作的工作代码，或者您是NURBS专家，否则我建议您使用一些NURBS库。通常，与您的问题相关的操作集合是:点评估、节点插入、分裂，以及可能的程度提升。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|397064|作为一般性，我将描述三种可能的解决方案。|397065|结裂|397066|假设您输入的NURBS曲线是非有理的(没有权值=单位权重)，并且它们的度不能超过所得到的Bezier曲线的最大允许度。然后样条的每个跨度都是一条多项式曲线，因此可以提取为Bezier曲线。|397067|根据您使用的库，算法的描述可能有所不同。以下是可能的变体：|397068|如果有一个函数将NURBS曲线分割成Bezier曲线，只需调用它。|ordered-list-item|397069|假设在给定的参数下，有一个将曲线分割成两个子曲线的函数。然后将你的曲线分裂成任何内部结(即不等于最小/最大节)。对每个子曲线做同样的操作，直到没有内部结，这意味着所有的曲线都是Bezier。|397070|任何NURBS库都必须具有节点插入功能。对于每个多重度小于D(度)的节点Ki，用param+=+Ki调用节点插入。您可以按任何顺序插入不同的结。库还可以包含“多个节点插入”，这允许将所有插入合并到一个调用中。最后，最小/最大节点必须具有多重性D%2B1，所有内部节点必须具有多重性D。此时控制点充分描述了所需的Bezier曲线:控制点0+.+D定义0-+Bezier，D.+2D定义第1-+Bezier，.，q.(q%2B1)+D控制点定义了q-th+Bezier.|397071|如果输入NURBS曲线的程度低于Bezier曲线所需的程度，您也可以对原始NURBS曲线或由此产生的Bezier曲线调用度提升。说到ID2D1GeometrySink，它接受所有<=+3度的贝塞尔曲线(线性贝塞尔曲线只是一个线段)，所以它是不必要的。|397072|如果你的NURBS曲线可能有不可接受的高度，或者是理性的，那么你必须用三次样条(更难更快)或者用折线(更简单但更慢)来逼近这条曲线。|397073|保折线逼近|397074|这里是一个相当简单的递归算法，它建立了具有保证误差<=+MaxErr的NURBS曲线的折线逼近。|397075|从曲线的第一个控制点到最后一个控制点画一个线段。|397076|检查是否所有控制点都在离段的MaxErr距离之内。|397077|如果是，则将线段添加到输出中。|397078|否则，将中间的曲线分割成两个子曲线，并递归地逼近它们。|397079|要实现它，您需要NURBS曲线分割操作(可以通过节点插入实现)。|397080|启发式折线逼近|397081|如果没有NURBS库在手边，实施结插入可能会造成很大的痛苦。这就是为什么我描述了另一个解决方案，它只使用NURBS曲线的点评估。您可以通过de算法或定义实现点计算(请参阅基函数和NURBS曲线定义)|offset|length|397082|该算法是递归的，它接受原始曲线上的一个参数区间作为输入。|397083|评估参数区间的起始点和终点。|397084|画一条线段穿过它们。|397085|求出参数区间内曲线上的若干点。|397086|检查这些内部点是否在线段的MaxErr距离内。|397087|397088|否则，将参数区间分成两半，然后递归地调用它们的逼近。|397089|该算法是自适应的，在一些罕见的情况下，在实际应用中会产生很差的逼近效果。检查点可以在参数区间内均匀选择。为了获得更高的可靠性，在参数区间内的输入曲线的所有节点处也要对曲线进行评估。|397090|第三方图书馆|397091|如果您不打算经常使用NURBS，我建议您使用锡样条库。它是非常小的设计，没有依赖，并拥有麻省理工学院的许可。而且，它似乎是积极开发的，因此您可以在遇到任何问题时与作者进行交流。|397092|似乎第一种解决方案对主题启动器来说已经足够了，下面是用微小样条将NURBS分割成Bezier曲线的代码：|397093|#include+<stdlib.h>
#include+<stdio.h>
#include+<assert.h>
#include+<math.h>
#include+"tinysplinecpp.h"
#include+"debugging.h"

int+main()+{
++++//create+B-spline+curve+and+set+its+data
++++TsBSpline+nurbs(3,+2,+10,+TS_NONE);
++++float+knotsData[]+=+{0.0f,+0.0f,+0.0f,+0.0f,+0.3f,+0.3f,+0.5f,+0.7f,+0.7f,+0.7f,+1.0f,+1.0f,+1.0f,+1.0f};
++++for+(int+i+=+0;+i+<+nurbs.nKnots();+i%2B%2B)
++++++++nurbs.knots()[i]+=+knotsData[i];
++++for+(int+i+=+0;+i+<+nurbs.nCtrlp();+i%2B%2B)+{
++++++++nurbs.ctrlp()[2*i%2B0]+=+0.0f+%2B+i;
++++++++float+x+=+1.0f+-+i+/+float(nurbs.nCtrlp());
++++++++nurbs.ctrlp()[2*i%2B1]+=+x+*+x+*+x;
++++}
++++ts_bspline_print(nurbs.data());

++++//insert+knots+into+B-spline+so+that+it+becomes+a+sequence+of+bezier+curves
++++TsBSpline+beziers+=+nurbs;
++++beziers.toBeziers();
++++ts_bspline_print(beziers.data());

++++//check+that+the+library+does+not+fail+us
++++for+(int+i+=+0;+i+<+10000;+i%2B%2B)+{
++++++++float+t+=+float(rand())+/+RAND_MAX;
++++++++float+*pRes1+=+nurbs(t).result();
++++++++float+*pRes2+=+beziers(t).result();
++++++++float+diff+=+hypotf(pRes1[0]+-+pRes2[0],+pRes1[1]+-+pRes2[1]);
++++++++if+(diff+>=+1e-6f)
++++++++++++printf("Bad+eval+at+%25f:+err+=+%25f++(%25f;%25f)+vs+(%25f;%25f)\n",+t,+diff,+pRes1[0],+pRes1[1],+pRes2[0],+pRes2[1]);
++++}

++++//extract+bezier+curves
++++assert(beziers.nCtrlp()+%25+nurbs.order()+==+0);
++++int+n+=+beziers.nCtrlp()+/+nurbs.order();
++++int+sz+=+nurbs.order()+*+2;+//floats+per+bezier
++++for+(int+i+=+0;+i+<+n;+i%2B%2B)+{
++++++++float+*begin+=+beziers.ctrlp()+%2B+sz+*+i;
++++++++float+*end+=+beziers.ctrlp()+%2B+sz+*+(i+%2B+1);
++++++++//[begin..end)+contains+control+points+of+i-th+bezier+curve
++++}

++++return+0;
}|code-block|syntax|javascript|397094|最后注|style|BOLD|397095|以上大多数文本都假设NURBS曲线是夹紧的，这意味着最小和最大节点具有多重D%2B1，有时也使用无夹紧NURBS曲线。如果您遇到一个，您还可能需要通过使用库的批准功能来夹紧它。方法toBeziers来自锡样条使用以上夹紧NURBS自动，您不需要手动夹紧它。|397096|entityMap|0|LINK|mutability|MUTABLE|url|http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-property.html|1|http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/NURBS-property.html|2|https://github.com/retuxx/tinyspline^0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|2D|3|0|2H|7|1|0|0|0|0|0|0|0|0|0|0|M|3|2|0|0|0|0|3|0|0^^$0|@$1|2|3|4|5|6|7|2M|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|2N|8|@]|9|@]|A|$]]|$1|D|3|E|5|6|7|2O|8|@]|9|@]|A|$]]|$1|F|3|G|5|6|7|2P|8|@]|9|@]|A|$]]|$1|H|3|I|5|6|7|2Q|8|@]|9|@]|A|$]]|$1|J|3|K|5|L|7|2R|8|@]|9|@]|A|$]]|$1|M|3|N|5|L|7|2S|8|@]|9|@]|A|$]]|$1|O|3|P|5|L|7|2T|8|@]|9|@]|A|$]]|$1|Q|3|R|5|6|7|2U|8|@]|9|@]|A|$]]|$1|S|3|T|5|6|7|2V|8|@]|9|@]|A|$]]|$1|U|3|V|5|6|7|2W|8|@]|9|@]|A|$]]|$1|W|3|X|5|6|7|2X|8|@]|9|@]|A|$]]|$1|Y|3|Z|5|L|7|2Y|8|@]|9|@]|A|$]]|$1|10|3|11|5|L|7|2Z|8|@]|9|@]|A|$]]|$1|12|3|13|5|L|7|30|8|@]|9|@]|A|$]]|$1|14|3|15|5|L|7|31|8|@]|9|@]|A|$]]|$1|16|3|17|5|6|7|32|8|@]|9|@]|A|$]]|$1|18|3|19|5|6|7|33|8|@]|9|@]|A|$]]|$1|1A|3|1B|5|6|7|34|8|@]|9|@$1C|35|1D|36|1|37]|$1C|38|1D|39|1|3A]]|A|$]]|$1|1E|3|1F|5|6|7|3B|8|@]|9|@]|A|$]]|$1|1G|3|1H|5|L|7|3C|8|@]|9|@]|A|$]]|$1|1I|3|1J|5|L|7|3D|8|@]|9|@]|A|$]]|$1|1K|3|1L|5|L|7|3E|8|@]|9|@]|A|$]]|$1|1M|3|1N|5|L|7|3F|8|@]|9|@]|A|$]]|$1|1O|3|13|5|L|7|3G|8|@]|9|@]|A|$]]|$1|1P|3|1Q|5|L|7|3H|8|@]|9|@]|A|$]]|$1|1R|3|1S|5|6|7|3I|8|@]|9|@]|A|$]]|$1|1T|3|1U|5|6|7|3J|8|@]|9|@]|A|$]]|$1|1V|3|1W|5|6|7|3K|8|@]|9|@$1C|3L|1D|3M|1|3N]]|A|$]]|$1|1X|3|1Y|5|6|7|3O|8|@]|9|@]|A|$]]|$1|1Z|3|20|5|21|7|3P|8|@]|9|@]|A|$22|23]]|$1|24|3|25|5|6|7|3Q|8|@$1C|3R|1D|3S|26|27]]|9|@]|A|$]]|$1|28|3|29|5|6|7|3T|8|@]|9|@]|A|$]]|$1|2A|3|-4|5|6|7|3U|8|@]|9|@]|A|$]]]|2B|$2C|$5|2D|2E|2F|A|$2G|2H]]|2I|$5|2D|2E|2F|A|$2G|2J]]|2K|$5|2D|2E|2F|A|$2G|2L]]]]

Unless you already have working code for basic NURBS operations or you are a NURBS expert, I would advise to use some NURBS library. Generally, the set of operations related to your problem are: point evaluation, knot insertion, splitting, and perhaps degree elevation.

For generality, I'll describe three possible solutions.

<h2>Split by knots</h2>

Suppose that your input NURBS curves are nonrational (no weights = unit weights), and their degree cannot exceed maximal allowed degree of resulting Bezier curves. Then each span of the spline is a polynomial curve, so it can be extracted as Bezier curve.

Depending on the library you use, the description of the algorithm can be different. Here are possible variants:

<ol>
<li>If there is a function to split a NURBS curve into Bezier curves, simply call it.</li>
<li>Suppose that there is a function for splitting a curve into two subcurves at given parameter. Then split your curve in any internal knot (i.e. not equal to min/max knots). Do the same for each of the subcurves until there are no internal knots, which means all the curves are Bezier.</li>
<li>Any NURBS library must have knot insertion function. For each knot Ki with multiplicity less than D (degree), call knot insertion with param = Ki. You can insert different knots in any order. The library can also contain "multiple knot insertion", which allows to combine all the insertions into one call. At the end, min/max knots must have multiplicity D+1, all the internal knots must have multiplicity D. At this moment the control points fully describe the Bezier curves you need: control points [0..D] define the 0-th Bezier, [D..2D] define the 1-th Bezier, ..., [q D .. (q+1) D] control points define the q-th Bezier.</li>
</ol>

If the degree of your input NURBS curve is lower that the required degree of Bezier curves, you can also call degree elevation either for the original NURBS curve, or for the resulting Bezier curves. Speaking of ID2D1GeometrySink, it accepts all Bezier curves with degree &lt;= 3 (linear Bezier curve is simply a line segment), so it is not necessary.

If your NURBS curve may have unacceptably high degree, or may be rational, then you have to approximate the curve with either cubic spline (harder and faster) or with polyline (simpler but slower).

<h2>Guaranteed polyline approximation</h2>

Here is a rather simple recursive algorithm that builds polyline approximation of a NURBS curve with guaranteed error &lt;= MaxErr.

<ol>
<li>Draw a line segment from the first to the last control points of the curve.</li>
<li>Check if all the control points are within MaxErr distance from the segment.</li>
<li>If they are, then add the line segment to the output.</li>
<li>Otherwise, split the curve at the middle into two subcurves, and approximate them recursively.</li>
</ol>

To implement it, you need NURBS curve splitting operation (which can be implemented via knot insertion).

<h2>Heuristic polyline approximation</h2>

If there is no NURBS library at hand, implementing knot insertion may cause a lot of pain. That's why I describe one more solution which uses only point evaluation of NURBS curves. You can implement point evaluation either via de Boor algorithm, or by definition (see <a href="http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-property.html" rel="noreferrer">basis functions</a> and <a href="http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/NURBS-property.html" rel="noreferrer">NURBS curve</a> definitions)

The algorithm is recursive, it accepts a parametric interval on the original curve as input.

<ol>
<li>Evaluate the start and the end points of the parametric interval.</li>
<li>Draw a line segment through them.</li>
<li>Evaluate some number of points on the curve inside the parametric interval.</li>
<li>Check that these internal points are within MaxErr distance of the line segment.</li>
<li>If they are, then add the line segment to the output.</li>
<li>Otherwise, split the parametric interval into two halves, and call approximation on them recursively.</li>
</ol>

This algorithm is adaptive, and it can produce bad approximation in practice in some rare cases. The check points can be chosen uniformly within the parametric interval. For more reliability, it's better to also evaluate the curve at all the knots of the input curve which fall within the parametric interval.

<h2>Third-party library</h2>

If you are not going to work with NURBS a lot, I suggest taking the <a href="https://github.com/retuxx/tinyspline" rel="noreferrer">tinyspline</a> library. It is very small by design, has no dependencies, and has MIT license. Also, it seems to be actively developed, so you can communicate to the author in case of any issues.

It seems that the first solution is enough for the topic starter, so here is the code for splitting NURBS into Bezier curves with tinyspline:

<pre><code>#include &lt;stdlib.h&gt;
#include &lt;stdio.h&gt;
#include &lt;assert.h&gt;
#include &lt;math.h&gt;
#include "tinysplinecpp.h"
#include "debugging.h"

int main() {
 //create B-spline curve and set its data
 TsBSpline nurbs(3, 2, 10, TS_NONE);
 float knotsData[] = {0.0f, 0.0f, 0.0f, 0.0f, 0.3f, 0.3f, 0.5f, 0.7f, 0.7f, 0.7f, 1.0f, 1.0f, 1.0f, 1.0f};
 for (int i = 0; i &lt; nurbs.nKnots(); i++)
 nurbs.knots()[i] = knotsData[i];
 for (int i = 0; i &lt; nurbs.nCtrlp(); i++) {
 nurbs.ctrlp()[2*i+0] = 0.0f + i;
 float x = 1.0f - i / float(nurbs.nCtrlp());
 nurbs.ctrlp()[2*i+1] = x * x * x;
 }
 ts_bspline_print(nurbs.data());

 //insert knots into B-spline so that it becomes a sequence of bezier curves
 TsBSpline beziers = nurbs;
 beziers.toBeziers();
 ts_bspline_print(beziers.data());

 //check that the library does not fail us
 for (int i = 0; i &lt; 10000; i++) {
 float t = float(rand()) / RAND_MAX;
 float *pRes1 = nurbs(t).result();
 float *pRes2 = beziers(t).result();
 float diff = hypotf(pRes1[0] - pRes2[0], pRes1[1] - pRes2[1]);
 if (diff &gt;= 1e-6f)
 printf("Bad eval at %f: err = %f (%f;%f) vs (%f;%f)\n", t, diff, pRes1[0], pRes1[1], pRes2[0], pRes2[1]);
 }

 //extract bezier curves
 assert(beziers.nCtrlp() % nurbs.order() == 0);
 int n = beziers.nCtrlp() / nurbs.order();
 int sz = nurbs.order() * 2; //floats per bezier
 for (int i = 0; i &lt; n; i++) {
 float *begin = beziers.ctrlp() + sz * i;
 float *end = beziers.ctrlp() + sz * (i + 1);
 //[begin..end) contains control points of i-th bezier curve
 }

 return 0;
}
</code></pre>

Final note

Most of the text above assumes that your NURBS curves are clamped, which means that min and max knots have multiplicity D+1. Unclamped NURBS curves are also used sometimes. If you meet one, you may also need to clamp it by using approproate function of the library. Method toBeziers from tinyspline used just above clamps NURBS automatically, you don't need to clamp it manually.

blocks|key|397135|text|我使用这个示例代码将基数样条转换为三次bezier补丁列表：http://www.codeproject.com/Articles/31859/Draw-a-Smooth-Curve-through-a-Set-of-D-Points-wit|type|unstyled|depth|inlineStyleRanges|entityRanges|offset|length|data|397136|它是为WPF编写的，但是由于WPF和Direct2D的编程模型不同(声明式和命令式)，所以很容易转换为Direct2D。|397137|entityMap|0|LINK|mutability|MUTABLE|url|http://www.codeproject.com/Articles/31859/Draw-a-Smooth-Curve-through-a-Set-of-D-Points-wit^0|U|2J|0|0|0^^$0|@$1|2|3|4|5|6|7|N|8|@]|9|@$A|O|B|P|1|Q]]|C|$]]|$1|D|3|E|5|6|7|R|8|@]|9|@]|C|$]]|$1|F|3|-4|5|6|7|S|8|@]|9|@]|C|$]]]|G|$H|$5|I|J|K|C|$L|M]]]]

I used this sample code for transforming a cardinal spline to a list of cubic bezier patches: <a href="http://www.codeproject.com/Articles/31859/Draw-a-Smooth-Curve-through-a-Set-of-D-Points-wit" rel="nofollow">http://www.codeproject.com/Articles/31859/Draw-a-Smooth-Curve-through-a-Set-of-D-Points-wit</a>

It's written for WPF, but since WPF and Direct2D differ only in their programming model (declarative vs. imperative), it translates very easily to Direct2D.

I have the data of a spline curve

<ul>
<li>Degree</li>
<li>Knots</li>
<li>Control points</li>
<li>Fit points</li>
</ul>

And I need to draw this curve by using <code>Direct2D</code>. At the moment I am using the <a href="https://msdn.microsoft.com/en-us/library/windows/desktop/dd316592%28v=vs.85%29.aspx">ID2D1GeometrySink interface</a> to draw geometries but it seems it does not implements a possible <code>AddSpline</code> method.

Is there a way to draw spline by means of <code>Direct2D</code>? Even a <code>DirectX</code> implementation that can be used in a <code>Direct2D</code> application will be fine.

Draw splines by using Direct2D

翻译质量差，导致语言生硬或混乱。

没有提供实际的解决方法或示例。

解答不清晰，无法理解或解决问题。

页面排版不美观，阅读体验差。

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我有样条曲线的数据学位结控制点契合点我需要用Direct2D绘制这条曲线。目前，我正在使用绘制几何图形，但它似乎没有实现一个可能的AddSpline方法。有没有一种用Direct2D绘制样条的方法？即使是可以在DirectX应用程序中使用的Direct2D实现也是可以的。

问用Direct2D绘制样条
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