|Drafting → Bézier tools → Cubic Bézier curve|
|Introduced in version|
|Draft BezCurve, Draft BSpline|
The Bézier Curve is one of the most commonly used curves in computer graphics. This command allows you to create a continuous spline made up of several 3rd-degree Bézier segments, in a way that is similar to the Bézier tool in Inkscape. A general Bézier curve of any degree can be created with the Draft BezCurve command.
The Draft BezCurve and the Draft CubicBezCurve commands use control points to define the position and curvature of the spline. The Draft BSpline command, on the other hand, specifies the exact points through which the curve will pass.
Spline defined by three cubic Bézier segments. Each cubic Bézier is defined by four points, but when the tool is used graphically, only three such points are placed: 1-2-3 for the first segment, 3-4-5 for the second segment, and 5-6-7 for the third segment; the fourth point in each segment is defined implicitly; the last point 8 is necessary to finish the operation, and would be part of a fourth Bézier segment if the operation is continued.
The following description is valid only for CAD Navigation. So first make sure that your mouse navigation is in correct mode.
There are several ways to invoke the command:
- Click the first point in the 3D view, and hold the mouse button (1); this is the first (starting) point.
- Drag the pointer to another point in the 3D view, and release the mouse button (2); this is the first control point.
- Move the pointer to another point in the 3D view, and click and hold the mouse button on this point (3); this is the second end point.
- Move the pointer to another point in the 3D view to adjust the final curvature of the spline, and then release the mouse button (4).
- At this moment you already have one Bézier curve of 3rd degree. The command can be completed by pressing or the button, or you may repeat the process of clicking and holding (5), and dragging and releasing (6) to add more 3rd-degree Bézier segments.
Note that with this workflow you need two click-hold-release sequences to create a single Bézier curve of third degree.
- The first click-hold defines the first end point.
- The first release defines the first control point.
- The second click-hold defines the second end point, and the general direction of the spline.
- The second release defines the final curvature of the spline.
- The second control point is not explicitly given, but is determined from the location of the pointer during the second release.
Creating several Bézier segments
- The second release also correspond to the first control point of the subsequent Bézier curve.
- This means that the second click-hold was also the first end point of the second Bézier curve.
- A third click-hold would be the second end point.
- A third release would define the final curvature of the second curve, and it would also be the first control point of a third curve.
This means that for every two click-hold (c-h) and release (r) sequences, the second sequence is already part of a subsequent curve segment, as indicated in the following graphic:
|c-h -- r -- c-h -- r| 1 |c-h -- r -- c-h -- r| 2 |c-h -- r -- c-h -- r| 3 |c-h -- r -- c-h -- r| 4
How to exactly place the control points
The graphical operation of this tool only allows the user to specify the first control point of the curve when it is being drawn. The second control point can be adjusted after the object is created: double click on the curve object in tree view, or select it and press. Then drag the second control point to the desired position.
In order to choose exactly both end points and both control points, the Python command
Draft.makeBezCurve() must be used. See the Scripting section.
See Draft BezCurve for general information. A cubic Bézier is created by passing the option
For each cubic Bézier segment four points must be used, of which the two extreme points indicate where the spline passes through, and the two intermediate points are control points.
- If only 3 points are given, it creates a quadratic Bézier instead, with only one control point.
- If only 2 points are given, it creates a linear Bézier, that is, a straight line.
- If 5 points are given, the first 4 create a cubic Bézier segment; the 4th and the 5th points are used to create a straight line.
- If 6 points are given, the first 4 create a cubic Bézier segment; the 4th and the other two points are used to create a quadratic Bézier segment.
- If 7 points are given, the first 4 create a cubic Bézier segment; the 4th and the other three points are used to create a second cubic Bézier segment.
- In general, the last point in a group of four is shared with the following three points maximum to create another Bézier segment.
- To have smooth curves, with no straight segments, the number of points should be
3n + 1or
nis the number of segments, for
n >= 1.
Examples of Bézier curves produced by using 2, 3, 4, 5, 6, 7, and 8 points. The solid lines indicate cubic Bézier segments; the other lines are quadratic or linear.
import FreeCAD as App import Draft p1 = App.Vector(-3500, 0, 0) p2 = App.Vector(-3000, 2000, 0) p3 = App.Vector(-1100, 2000, 0) p4 = App.Vector(0, 0, 0) p5 = App.Vector(1500, -2000, 0) p6 = App.Vector(3000, -1500, 0) p7 = App.Vector(5000, 0, 0) p8 = App.Vector(6000, 1500, 0) rot = App.Rotation() c1 = Draft.make_circle(100, placement=App.Placement(p1, rot), face=False) c1.Label = "B1_E1" c2 = Draft.make_circle(50, placement=App.Placement(p2, rot), face=True) c2.Label = "B1_c1" c3 = Draft.make_circle(50, placement=App.Placement(p3, rot), face=True) c3.Label = "B1_c2" c4 = Draft.make_circle(100, placement=App.Placement(p4, rot), face=False) c4.Label = "B1_E2" c5 = Draft.make_circle(50, placement=App.Placement(p5, rot), face=True) c5.Label = "B2_c3" c6 = Draft.make_circle(50, placement=App.Placement(p6, rot), face=True) c6.Label = "B2_c4" c7 = Draft.make_circle(100, placement=App.Placement(p7, rot), face=False) c7.Label = "B2_E3" c8 = Draft.make_circle(50, placement=App.Placement(p8, rot), face=True) c8.Label = "B3_c5" App.ActiveDocument.recompute() B1 = Draft.make_bezcurve([p1, p2], degree=3) B1.Label = "B_lin" B1.ViewObject.DrawStyle = "Dashed" B2 = Draft.make_bezcurve([p1, p2, p3], degree=3) B2.Label = "B_quad" B2.ViewObject.DrawStyle = "Dotted" B3 = Draft.make_bezcurve([p1, p2, p3, p4], degree=3) B3.Label = "B_cub" B3.ViewObject.LineWidth = 4 B4 = Draft.make_bezcurve([p1, p2, p3, p4, p5], degree=3) B4.Label = "B_cub+lin" B4.ViewObject.DrawStyle = "Dashed" B5 = Draft.make_bezcurve([p1, p2, p3, p4, p5, p6], degree=3) B5.Label = "B_cub+quad" B5.ViewObject.DrawStyle = "Dotted" B6 = Draft.make_bezcurve([p1, p2, p3, p4, p5, p6, p7], degree=3) B6.Label = "B_cub+cub" B6.ViewObject.LineWidth = 2 B7 = Draft.make_bezcurve([p1, p2, p3, p4, p5, p6, p7, p8], degree=3) B7.Label = "B_cub+cub+lin" B7.ViewObject.DrawStyle = "Dashed" App.ActiveDocument.recompute()