From e5f598ff5c868efecace80a05ae844c756bae437 Mon Sep 17 00:00:00 2001 From: Juan Linietsky Date: Tue, 23 Apr 2019 14:14:46 -0300 Subject: [PATCH] Add tutorials on interpolation and beziers. --- tutorials/math/beziers_and_curves.rst | 178 ++++++++++++++++++ tutorials/math/img/bezier_cubic_handles.png | Bin 0 -> 12685 bytes tutorials/math/img/bezier_cubic_points.gif | Bin 0 -> 216264 bytes tutorials/math/img/bezier_cubic_points.png | Bin 0 -> 7687 bytes .../math/img/bezier_interpolation_baked.gif | Bin 0 -> 51759 bytes .../math/img/bezier_interpolation_speed.gif | Bin 0 -> 55424 bytes tutorials/math/img/bezier_path_2d.png | Bin 0 -> 21289 bytes tutorials/math/img/bezier_point_amount.png | Bin 0 -> 4692 bytes .../math/img/bezier_quadratic_points.png | Bin 0 -> 6674 bytes .../math/img/bezier_quadratic_points2.gif | Bin 0 -> 153902 bytes tutorials/math/img/interpolation_follow.gif | Bin 0 -> 110634 bytes tutorials/math/img/interpolation_monkey.gif | Bin 0 -> 285232 bytes .../math/img/interpolation_positions.png | Bin 0 -> 6592 bytes tutorials/math/img/interpolation_vector.gif | Bin 0 -> 21734 bytes tutorials/math/index.rst | 2 + tutorials/math/interpolation.rst | 105 +++++++++++ 16 files changed, 285 insertions(+) create mode 100644 tutorials/math/beziers_and_curves.rst create mode 100644 tutorials/math/img/bezier_cubic_handles.png create mode 100644 tutorials/math/img/bezier_cubic_points.gif create mode 100644 tutorials/math/img/bezier_cubic_points.png create mode 100644 tutorials/math/img/bezier_interpolation_baked.gif create mode 100644 tutorials/math/img/bezier_interpolation_speed.gif create mode 100644 tutorials/math/img/bezier_path_2d.png create mode 100644 tutorials/math/img/bezier_point_amount.png create mode 100644 tutorials/math/img/bezier_quadratic_points.png create mode 100644 tutorials/math/img/bezier_quadratic_points2.gif create mode 100644 tutorials/math/img/interpolation_follow.gif create mode 100644 tutorials/math/img/interpolation_monkey.gif create mode 100644 tutorials/math/img/interpolation_positions.png create mode 100644 tutorials/math/img/interpolation_vector.gif create mode 100644 tutorials/math/interpolation.rst diff --git a/tutorials/math/beziers_and_curves.rst b/tutorials/math/beziers_and_curves.rst new file mode 100644 index 000000000..ae23febaa --- /dev/null +++ b/tutorials/math/beziers_and_curves.rst @@ -0,0 +1,178 @@ +.. _doc_beziers_and_curves: + +Beziers, Curves and Paths +========================== + +Introduction +~~~~~~~~~~~~ + +Bezier curves are a mathematical approximation to natural shapes. Their goal is to represent a curve with +as little information as possible, and with a high level of flexibility. + +Unlike other more abstract mathematical concepts, Bezier curves were created for industrial design and are +widely popular in the graphics software industry. + +The way they work is very simple, but to understand it, let's start from the most minimal example. + + +Quadratic Bezier +---------------- + +Take three points (the minimum required): + +.. image:: img/bezier_quadratic_points.png + +To draw the curve between them, just interpolate the two segments that form between the three points, individually (using values 0 to 1). This will result in two points. + +.. tabs:: + .. code-tab:: gdscript GDScript + + func _quadratic_bezier(p0 : Vector2,p1 : Vector2,p2 : Vector2 ,t : float): + var q0 = p0.linear_interpolate(p1,t) + var q1 = p1.linear_interpolate(p2,t) + +This will reduce the points from 3 to 2. Do the same process with *q0* and *q1* to obtain a single point *r*. + +.. tabs:: + .. code-tab:: gdscript GDScript + + var r = q0.linear_interpolate(q1,t) + return r + +Finally, this point fill follow the curve when t goes from 0 to 1. This type of curve is called *Quadratic Bezier*. + +.. image:: img/bezier_quadratic_points2.gif + +*(Image credit: Wikipedia)* + +Cubic Bezier +---------------- + +Let's add one more point and make it four. + +.. image:: img/bezier_cubic_points.png + +Then let's modify the function to take four points as an input, *p0, p1, p2* and *p3*: + +.. tabs:: + .. code-tab:: gdscript GDScript + + func _cubic_bezier(p0 : Vector2,p1 : Vector2,p2 : Vector2,p3 : Vector2 ,t : float): + +Interpolate then into three points: + +.. tabs:: + .. code-tab:: gdscript GDScript + + var q0 = p0.linear_interpolate(p1,t) + var q1 = p1.linear_interpolate(p2,t) + var q2 = p2.linear_interpolate(p3,t) + +From there to two points: + +.. tabs:: + .. code-tab:: gdscript GDScript + + var r0 = q0.linear_interpolate(q1,t) + var r1 = q1.linear_interpolate(q2,t) + +And to one: + +.. tabs:: + .. code-tab:: gdscript GDScript + + var s = r0.linear_interpolate(r1,t) + return s + +The result will be a smooth curve interpolating between all four points: + +.. image:: img/bezier_quadratic_points2.gif + +*(Image credit: Wikipedia)* + +.. note:: For 3D, it's exactly the same, just change Vector2 into Vector3. + +Control point form +------------------- + +Now, let's take these points and change the way we understand them. Instead of having p0, p1, p2 and p3, we will store them as: + +* **POINT0** = **P0**: Is the first point, the source +* **CONTROL0** = **P1** - **P0**: Is a relative vector for the first control point +* **CONTROL1** = **P3** - **P2**: Is a relative vector for the second control point +* **POINT1** = **P3**: Is the second point, the destination + +This way, we have two points and two control points (which are relative vectors to the respective points). If visualized, they will look a lot more familiar: + +.. image:: img/bezier_cubic_handles.png + +This is actually how graphics software presents Bezier curves to the users, and how Godot supports them. + +Curve, Curve2D, Path and Path2D +------------------------------- + +There are two objects that contain curves: :ref:`Curve ` and :ref:`Curve ` (for 3D and 2D respectively). + +They can contain several points, allowing for longer paths. It is also possible to set them to nodes: :ref:`Path ` and :ref:`Path2D ` (also for 3D and 2D respectively): + +.. image:: img/bezier_path_2d.png + +Using them, however, may not be completely obvious, so following is a description of the most common use cases for Bezier curves. + +Evaluating +----------- + +Just evaluating them may be an option, but in most cases it's not very useful. The big drawback with Bezier curves is that if you traverse them at constant speed, from *t=0* to *t=1*, the actual interpolation will *not* move at constant speed. The speed is also an interpolation between the distances between points p0, p1, p2 and p3 and there is not a mathematically simple way to traverse the curve at constant speed. + +Let's do a simple example with the following pseudocode: + +.. tabs:: + .. code-tab:: gdscript GDScript + + var t = 0.0 + _process(delta): + t+=delta + position = _cubic_bezier(p0,p1,p2,p3,t) + + +.. image:: img/bezier_interpolation_speed.gif + +As you can see, the speed (in pixels per second) of the circle varies, even though *t* is increased at constant speed. This makes beziers difficult to use for anything practical out of the box. + +Drawing +------- + +Drawing beziers (or objects based on the curve) is a very common use case, but it's also not easy. For pretty much any case, Bezier curves need to be converted to some sort of segments. This is normally difficult, however, without creating a very high amount of them. + +The reason is that some sections of a curve (specifically, corners) may requiere considerably points, while other sections may not: + +.. image:: img/bezier_point_amount.png + +Additionally, if both control points were 0,0 (remember they are relative vectors), the Bezier curve would just be a straight line (so drawing a high amount of points would be wasteful). + +Before drawing Bezier curves, *tesselation* is required. This is often done with a recursive or divide and conquer function that splits the curve until the curvature amount becomes less than a certain threshold. + +The *Curve* classes provide this via the :ref:`Curve.tesselate()` function (which receives optional *stages* of recursion and angle *tolerance* arguments). This way, drawing something based on a curve is easier. + +Traversal +--------- + +The last common use case for the curves is to traverse them. Because of what was mentioned before regarding constant speed, this is also difficult. + +To make this easier, the curves need to be *baked* into equidistant points. This way, they can be approximated with regular interpolation (which can be improved further with a cubic option). To do this, just use the :ref:`Curve.interpolate_baked()` method together with :ref:`Curve.get_baked_length()`. The first call to either of them will bake the curve internally. + +Traversal at constant speed, then, can be done with the following pseudo-code: + +.. tabs:: + .. code-tab:: gdscript GDScript + + var t = 0.0 + _process(delta): + t+=delta + position = curve.interpolate_baked( t * curve.get_baked_length(), true) + +And the output will, then, move at constant speed: + +.. image:: img/bezier_interpolation_baked.gif + + diff --git a/tutorials/math/img/bezier_cubic_handles.png b/tutorials/math/img/bezier_cubic_handles.png new file mode 100644 index 0000000000000000000000000000000000000000..c587edac50dc07208eec4aafc4613c2598542d9a GIT binary patch literal 12685 zcmZX*byQT}7dOnz(B0h(EfSK_$WRi3NS82lH%QG0B16sqQWA zzBkhR!aZUMf>;DS@%8q0_6fp!65#wY$QjBV>K^3At*vYD%p6Y1gonqCr>m)M8aBII z5Sl{|xf)RIY(V8_kpt858Qy3akI-~Jk<3?!u#?dKiT5he_3Y1a+28QQO%@3*7l^VX zj|sjc&r924;?@#p(AP(|*Q-^i#X`<$TiMaQ>+r&G#`cI;F8@{hv);G%?IcJn7W%)A z6e{f`HDgvV%?Ao3fFAT7W|4T86@(?eW^#}`@E#0qJ3IFwlyzfpn=Juxz<#P#&@)wl 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+ +Vector types (Vector2 and Vector3) can also be interpolated, they come with handy functions to do it +:ref:`Vector2.linear_interpolate()` and :ref:`Vector3.linear_interpolate()`. + +For cubic interpolation, there are also :ref:`Vector2.cubic_interpolate()` and :ref:`Vector3.cubic_interpolate()`, which do a :ref:`Bezier ` style interpolation. + +Here is simple pseudo-code in going from point A to B using interpolation: + +.. tabs:: + .. code-tab:: gdscript GDScript + + func _physics_process(delta): + + t+=delta*0.4 + $Sprite.position = $A.position.linear_interpolate( $B.position,t ) + +It will produce the following motion: + +.. image:: img/interpolation_vector.gif + +Transform interpolation +-------------------- + +It is also possible to interpolate whole transforms (make sure they have either uniform scale or, at least, the same non-uniform scale). +For this, the function :ref:`Transform.interpolate_with()` can be used. + +Here is an example of transforming a monkey from Position1 to Position2: + +.. image:: img/interpolation_positions.png + +Using the following pseudocode: + +.. tabs:: + .. code-tab:: gdscript GDScript + + var t = 0.0 + + func _process(delta): + + t+=delta + + $Monkey.transform = $Position1.transform.interpolate_with( $Position2.transform, t ) + +And again, it will produce the following motion: + +.. image:: img/interpolation_monkey.gif + + +Smoothing Motion +---------------- + +Interpolation can be used to smooth movement, rotation, etc. Here is an example of a circle following the mouse using smoothed motion: + +.. tabs:: + .. code-tab:: gdscript GDScript + + const FOLLOW_SPEED = 4.0 + + func _physics_process(delta): + + var mouse_pos = get_local_mouse_position() + + $Sprite.position = $Sprite.position.linear_interpolate( mouse_pos, delta * FOLLOW_SPEED ) + +Here is how it looks: + +.. image:: img/interpolation_follow.gif + +This useful for smoothing camera movement, allies following you (ensuring they stay within a certain range), and many other common game patterns. + +

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