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<title>Ipe Manual -- 5.1 Path objects</title>
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<table width="100%" cellpadding=0 cellspacing=2><tr><td bgcolor="#99ccff"><a href="manual_17.html"><img border="0" alt="5.2 Text objects" src="next.png"></a></td><td bgcolor="#99ccff"><a href="manual_15.html"><img border="0" alt="5 Object types" src="up.png"></a></td><td bgcolor="#99ccff"><img alt="" src="blank.png"></td><td align="center" bgcolor="#99ccff" width="100%"><b>5.1 Path objects</b></td></tr></table>
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<h2>5.1 Path objects</h2>
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Path objects are defined by a set of <em>subpaths</em>, that is, curves
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in the plane.<sup><a href="manual_44.html#id4">4</a></sup> Each subpath is
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either open or closed, and consists of straight line segments,
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circular or elliptic arc segments, parabola segments, and cubic
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B-spline segments. The curves are drawn with the stroke color, dash
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style, and line width; the interior of the object specified is filled
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<p>The distinction between open and closed subpaths is meaningful for
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stroking only, for filling any open subpath is implicitely closed.
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Stroking a set of subpaths is identical to stroking them individually.
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This is not true for filling: using several subpaths, one can
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construct objects with holes, and more complicated pattern. The
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filling algorithm is the <em>even-odd rule</em> of Postscript/PDF: To
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determine whether a point lies inside the filled shape, draw a ray
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from that point in any direction, and count the number of path
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segments that cross the ray. If this number is odd, the point is
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inside; if even, the point is outside.
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<p>Ipe can draw arrows on the first and last segment of a path object,
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but only if that segment is part of an open subpath.
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<p>There are several Ipe modes that create path objects in different
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ways. All modes create an object consisting of a single subpath
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only. To make more complicated path objects, such as objects with
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holes, you create each boundary component separately, then select them
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all and use the <em>Compose paths</em> function in the <em>Edit</em> menu.
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The reverse operation is <em>Decompose path</em>, you find it in the
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object menu of a path object that has several subpaths.
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<p>You can also create complicated paths by joining curves sequentially.
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For this to work, the endpoint of one path must be (nearly) identical
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to the begin point of the next--easy to achieve using snapping. You
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select the object you wish to join, and call <em>Join paths</em> in the
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<em>Edit</em> menu. You can also join several open path objects into a
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single closed path this way.
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Circles</A> can be entered in three different ways. To create an ellipse,
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create a circle and stretch and rotate it. Circular arcs can be
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entered by clicking three points on the arc or by clicking the center
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of the supporting circle as well as the begin and end vertex of the
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arc. They can be filled in Postscript fashion, and can have
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arrows. You can stretch a circular arc to create an elliptic arc.
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<p>A common application for arcs is to mark angles in drawings. The snap
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keys are useful to create such arcs: set arc creation to <em>center
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& 2 pts</em>, select <em>snap to vertex</em> and <em>snap to boundary</em>,
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click first on the center point of the angle (which is magnetic) and
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click then on the two bounding lines.
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There</A> are two modes for creating more complex general path objects.
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The difference between the <em>line</em> mode and the <em>polygon</em>
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mode is that the first creates an open path, the latter generates a
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closed one. As a consequence, the <em>line</em> mode uses the current
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arrow settings, while the <em>polygon</em> mode doesn't. Also, objects
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created in <em>line</em> mode will only have the stroke color set, even
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if the current fill color is not void. (However, if the current
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stroke color is void and the fill color is not, it will be filled.)
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<p>The path created consists of segments of various types. The initial
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setting is to create straight segments. By holding the shift-key when
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pressing the left mouse button one can switch to uniform
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B-splines. One can also add parabolic segments and circular arcs to
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the path by pressing the "q" and "a" keys.
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The mode for splines is identical to the polyline mode, but starts in
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the uniform B-spline setting.
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<p>For the mathematically inclined, a more precise description of the
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segments that can appear on a subpath follows. More details can be
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found in Foley et al.<sup><a href="manual_44.html#id5">5</a></sup> and other text books on splines.
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<p>A subpath consists of a sequence of segments. Each segment is either
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a straight line segment, an elliptic arc, a quadratic Bézier spline,
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a cubic Bézier spline, or a uniform cubic B-spline.
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<p>The quadratic Beziér spline defined by control points <i>p<sub>0</sub></i>,
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<i>p<sub>1</sub></i>, and <i>p<sub>2</sub></i>, is the curve
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P(t) = (1-t)<sup>2</sup> p<sub>0</sub> + 2t(1-t) p<sub>1</sub> + t<sup>2</sup> p<sub>2</sub>,
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where <i>t</i> ranges from <i>0</i> to <i>1</i>. This implies that it starts in
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<i>p<sub>0</sub></i> tangent to the line <i>p<sub>0</sub>p<sub>1</sub></i>, ends in <i>p<sub>2</sub></i> tangent to
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the line <i>p<sub>1</sub>p<sub>2</sub></i>, and is contained in the convex hull of the
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three points. Any segment of any parabola can be expressed as a
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quadratic Beziér spline.
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<p>For instance, the piece of the unit parabola <i>y = x<sup>2</sup></i> between <i>x=a</i>
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and <i>x=b</i> can be created with the control points
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<blockquote><table border><tbody><tr><td colspan="1" align="LEFT">
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p<sub>0</sub> </td><td colspan="1" align="CENTER"> = </td><td colspan="1" align="LEFT"> (a, a<sup>2</sup>)</td></tr>
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<tr><td colspan="1" align="LEFT">
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p<sub>1</sub> </td><td colspan="1" align="CENTER"> = </td><td colspan="1" align="LEFT"> ( (a + b)/(2), ab) </td></tr>
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<tr><td colspan="1" align="LEFT">
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p<sub>2</sub> </td><td colspan="1" align="CENTER"> = </td><td colspan="1" align="LEFT"> (b, b<sup>2</sup>)
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</i></td></tr></tbody></table>
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Any piece of any parabola can be created by applying some affine
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transformation to these points.
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<p>The cubic Beziér spline with control points <i>p<sub>0</sub></i>, <i>p<sub>1</sub></i>, and
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<i>p<sub>2</sub></i> is the curve
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R(t) = (1-t)<sup>3</sup>p<sub>0</sub> + 3t(1-t)<sup>2</sup> p<sub>1</sub> +
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3 t<sup>2</sup>(1-t) p<sub>2</sub> + t<sup>3</sup> p<sub>3</sub>.
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It starts in <i>p<sub>0</sub></i> being tangent to the line <i>p<sub>0</sub>p<sub>1</sub></i>, ends in
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<i>p<sub>3</sub></i> being tangent to the line <i>p<sub>2</sub>p<sub>3</sub></i>, and lies in the convex
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hull of the four control points.
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<p>Uniform cubic B-splines approximate a series of <i>m+1</i> control points
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<i>p<sub>0</sub></i>, <i>p<sub>1</sub></i>, ..., <i>p<sub>m</sub></i>, <i>m >= 3</i>, with a curve consisting
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of <i>m-2</i> cubic polynomial curve segments <i>s<sub>0</sub></i>, <i>s<sub>1</sub></i>, ...,
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<i>s<sub>m-3</sub></i>. Every such curve segment is defined by four of the control
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points. In fact, curve segment <i>s<sub>i</sub></i> is defined by the points
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<i>p<sub>i</sub></i>, <i>p<sub>i+1</sub></i>, <i>p<sub>i+2</sub></i>, and <i>p<sub>i+3</sub></i>. If the curve is closed
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(a <em>splinegon</em>), it contains three additional curve segments
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<i>s<sub>m-2</sub></i>, <i>s<sub>m-1</sub></i>, and <i>s<sub>m</sub></i>, defined by appending <i>p<sub>0</sub></i>,
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<i>p<sub>1</sub></i>, and <i>p<sub>2</sub></i> to the end of the sequence. A uniform B-spline
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segment on an open subpath of an Ipe path object is defined by
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repeating both the first and last control point three times, so as to
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make the segment begin and end in these points.
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<p>The segment <i>s<sub>i</sub></i> is the cubic curve segment with the following
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Q(t) = ((1-t)<sup>3</sup>)/(6) p<sub>i</sub> + (3t<sup>3</sup> -6t<sup>2</sup> + 4)/(6) p<sub>i+1</sub>
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+ (-3t<sup>3</sup> + 3t<sup>2</sup> + 3t +1)/(6) p<sub>i+2</sub> + (t<sup>3</sup>)/(6)
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where <i>t</i> ranges from <i>0</i> to <i>1</i>.
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<p>Since the point <i>Q(t)</i> is a convex combination of the four control
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points, the curve segment <i>s<sub>i</sub></i> lies in the convex hull of <i>p<sub>i</sub></i> to
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<i>p<sub>i+3</sub></i>. Furthermore, it follows that any affine transformation can be
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applied to the curve by applying it to the control points.
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Note that a control point <i>p<sub>i</sub></i> has influence on only four curve
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segments, <i>s<sub>i-3</sub></i>, <i>s<sub>i-2</sub></i>, <i>s<sub>i-1</sub></i>, and <i>s<sub>i</sub></i>. Thus, when you
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edit a spline object and move a control point, only a short piece of the
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spline in the neighborhood of the control point will move.
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<table width="100%" cellpadding=0 cellspacing=2><tr><td bgcolor="#99ccff"><a href="manual_17.html"><img border="0" alt="5.2 Text objects" src="next.png"></a></td><td bgcolor="#99ccff"><a href="manual_15.html"><img border="0" alt="5 Object types" src="up.png"></a></td><td bgcolor="#99ccff"><img alt="" src="blank.png"></td><td align="center" bgcolor="#99ccff" width="100%"><b>5.1 Path objects</b></td></tr></table></body></html>
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