Dev:Bone Layer - Revision history

Zelgadis: moved Bone Layer to Dev:Bone Layer

2010-02-19T07:04:40Z

moved Bone Layer to Dev:Bone Layer

Zelgadis: Text replace - ']]' to '}}'

2009-09-08T06:30:50Z

Text replace - ']]' to '}}'

Zelgadis: Text replace - '[[' to '{{l|'

2009-09-07T15:20:43Z

Text replace - '[[' to '{{l|'

Genete: /* Calculate the position and angle of a bone in a skeleton */ Same change here

2008-12-16T23:24:21Z

‎Calculate the position and angle of a bone in a skeleton: Same change here

Genete: /* Position of a Point due to Bones Influence */ Oops I reversed the matrix multiplications! thanks dooglus!

2008-12-16T23:07:21Z

‎Position of a Point due to Bones Influence: Oops I reversed the matrix multiplications! thanks dooglus!

Dooglus: /* What happen if there are more than one bone affecting the point? */

2008-11-30T11:30:51Z

‎What happen if there are more than one bone affecting the point?

Dooglus: /* Can Points (Bone influenced) be moved independently? */

2008-11-30T11:27:50Z

‎Can Points (Bone influenced) be moved independently?

Dooglus: /* Can Points (Bone influenced) be moved independently? */

2008-11-30T11:26:52Z

‎Can Points (Bone influenced) be moved independently?

Dooglus: /* How does a Vertex move under the influence of a single Bone? */

2008-11-30T11:19:33Z

‎How does a Vertex move under the influence of a single Bone?

Genete: /* Setup and Animation */

2008-11-30T10:53:59Z

‎Setup and Animation

@@ Line 44: / Line 44: @@
 Conceptually a bone has influence over a point by doing [http://en.wikipedia.org/wiki/Affine_transformation affine transformations] of its position according to the bone values. See also [http://en.wikipedia.org/wiki/Transformation_matrix transformation matrix]. A bone produces rotation, scale and translation to a point in this way:
-{{l|Image:Bonesimulation2.png]]
+{{l|Image:Bonesimulation2.png}}
 As you can see the values that produce the movement of the point P to the point P' are given by:
@@ Line 132: / Line 132: @@
 For those IK and FK calculations it is good to have the bone referenced to its parent. So the parameters of the bone are described as relative to the parent bone. The parent bone defines a local coordinate system where the origin of the coordinate system lies on the Origin of the bone and the tip of the bone is pointing to the positive direction of the X axis.
-{{l|Image: skeleton.png]]
+{{l|Image: skeleton.png}}
 As you can see in this image, each skeleton position and orientation is defined by the relative position of its immediate parent bone. In the diagram bone3 is a child of bone2 and bone 2 is a child of bone1. The bone 1 is the root parent bone but you can imagine it attached to a "master root bone" (defined by the origin of coordinates -red circumference- and the horizontal) so it can be said that the bone1 has its origin and angle defined relative to its "parent".

@@ Line 44: / Line 44: @@
 Conceptually a bone has influence over a point by doing [http://en.wikipedia.org/wiki/Affine_transformation affine transformations] of its position according to the bone values. See also [http://en.wikipedia.org/wiki/Transformation_matrix transformation matrix]. A bone produces rotation, scale and translation to a point in this way:
-[[Image:Bonesimulation2.png]]
+{{l|Image:Bonesimulation2.png]]
 As you can see the values that produce the movement of the point P to the point P' are given by:
@@ Line 132: / Line 132: @@
 For those IK and FK calculations it is good to have the bone referenced to its parent. So the parameters of the bone are described as relative to the parent bone. The parent bone defines a local coordinate system where the origin of the coordinate system lies on the Origin of the bone and the tip of the bone is pointing to the positive direction of the X axis.
-[[Image: skeleton.png]]
+{{l|Image: skeleton.png]]
 As you can see in this image, each skeleton position and orientation is defined by the relative position of its immediate parent bone. In the diagram bone3 is a child of bone2 and bone 2 is a child of bone1. The bone 1 is the root parent bone but you can imagine it attached to a "master root bone" (defined by the origin of coordinates -red circumference- and the horizontal) so it can be said that the bone1 has its origin and angle defined relative to its "parent".

@@ Line 230: / Line 230: @@
 <math>Q^0_j(t) = O^0_j(t) \cdot B^0_j(t)</math>
-<math>\beta^0_j(t) = \sum_{i=j}^{root} \alpha^0_i(t)</math>
+<math>\beta^0_j(t) = \sum_{i=root}^{j} \alpha^0_i(t)</math>
 where
-<math>B^0_j(t)=\prod_{i=parent}^{root} B^0_i(t)</math>
+<math>B^0_j(t)=\prod_{i=root}^{parent} B^0_i(t)</math>
-where 'i' goes form the parent of the bone 'j' and climbs up by one parent each time until it reaches the 'root' bone of the skeleton.
+where 'i' goes form the root bone to the parent of the bone 'j' and climbs down by one child each time until it reaches the 'parent of bone j' bone of the skeleton.
 <math>B^0_i(t)= T(-O^0_i(t)) \cdot R(-\alpha^0_i(t)) \cdot S_x(\frac{1}{s^0_j})</math>
@@ Line 250: / Line 250: @@
 <math>Q_j(t) = O^0_j(t) \cdot B^0_j(t) \cdot B_j(t)</math>
-<math>\beta_j(t)= \beta^0_j(t) + \sum_{i=root}^{parent} \alpha_i(t)</math>
+<math>\beta_j(t)= \beta^0_j(t) + \sum_{i=parent}^{root} \alpha_i(t)</math>
-where <math>B_j(t)=\prod_{i=root}^{parent} B_i(t)</math>
+where <math>B_j(t)=\prod_{i=parent}^{root} B_i(t)</math>
-where 'i' goes form the root bone and jumps down by one child each time until it reaches the 'parent' bone of the bone 'j'.
+where 'i' goes form the parent bone and jumps up by one parent each time until it reaches the 'root' bone of the bone 'j'.
 <math>B_i(t)= S_x(s_j(t)) \cdot R(\alpha_i(t)) \cdot T(O_i(t)) </math>

@@ Line 207: / Line 207: @@
 where i is a index that runs all the Bones Influence ID from i=1 to N (N= total number of bones that influence to the point)
-<math> B^0_i(t)= \prod_{j=bone}^{j=root} C_j^0(t)</math>
+<math> B^0_i(t)= \prod_{j=root}^{j=i} C_j^0(t)</math>
-<math> B_i(t)= \prod_{j=root}^{j=bone} C_j(t)</math>
+<math> B_i(t)= \prod_{j=i}^{j=root} C_j(t)</math>
-where 'j' runs the bone skeleton jumping from a child/parent bone to a parent/child bone each time. The travel of the index 'j' from ''bone'' to ''root'' and from ''root'' to ''bone'' is fixed -and unique- by the nature of skeletons constructions (a bone, a parent), as mentioned before.
+where 'j' runs the bone skeleton jumping from a child/parent bone to a parent/child bone each time. The travel of the index 'j' from ''root'' to ''i'' and from ''i'' to ''root'' is fixed -and unique- by the nature of skeletons constructions (a bone, a parent), as mentioned before.
 <math> C_j^0(t) = T(-O_j^0(t)) \cdot R(-\alpha_j^0(t)) \cdot S_x(\frac{1}{s^0_j})</math>

@@ Line 114: / Line 114: @@
 ===What happen if there are more than one bone affecting the point?===
-The first thing you can think when you deal with more than one bone making influence over the same point is to calculate the average of the sum of the influences of all bones. So if there are N bones affecting to the point then we can think on sum all the resulting points from each bone and then divide the result by N to not scale the result.
+The first thing you can think when you deal with more than one bone having influence over the same point is to calculate the average of the sum of the influences of all bones. So if there are N bones affecting the point then we can think of summing all the resulting points from each bone and then dividing the result by N to not scale the result.
 <math> B=\frac{\sum_{k=1}^N B_k}{N} </math>
@@ Line 120: / Line 120: @@
 Where <math>B_k</math> is the bone matrix for the bone ''k'' on the point P.
-That's fine but I would like to have more control to the way the bones affects to the points. When you move a bone it looks reasonable that not all the points are influence in the same way. This can be solved by using "weighted" average instead of a plain one. That means give different amount of influence to one bone than to other.
+That's fine but I would like to have more control to the way the bones affect the points. When you move a bone it looks reasonable that not all the points are influenced in the same way. This can be solved by using a "weighted" average instead of a plain one. That means giving a different amount of influence to one bone than to the other.
 <math> B=\frac{\sum_{k=1}^N B_k \cdot w_k}{\sum_{k=1}^N w_k} </math>

← Older revision	Revision as of 07:04, 19 February 2010
(No difference)

@@ Line 109: / Line 109: @@
 #In any position of the time line (bones are animated) the point has a position given by the influence of the bones: P(B).
-#The user wrap a point, drags it, placing it at P'.
+#The user selects a point, drags it, placing it at P'.
 #The final position of the point is P' but the user cannot modify the first term P(B) so the second term (b) becomes: P(animated)=P'-P(B).

@@ Line 102: / Line 102: @@
 === Can Points (Bone influenced) be moved independently? ===
-Yes, they should be. If not the animator is tight to only do the animation with the bones animation what would end to some insoluble problems during the animation. Considered this, the point position result comes from the addition of two terms:
+Yes, they should be. If not the animator is restricted to only doing the animation with bones which would lead to some insoluble problems during the animation. Considering this, the point position result comes from the addition of two terms:
 *(a) The movement of the point as described by the influence of the bone(s): P(B)
 *(b) The free movement of the point in the 2D space. P(animated)
-In terms of the interface with the user, the value P(animated)is just obtained in this way:
+In terms of the interface with the user, the value P(animated) is just obtained in this way:
-#In any position of the time line (bones are animated) the point have a position given by the influence of the bones: P(B).
+#In any position of the time line (bones are animated) the point has a position given by the influence of the bones: P(B).
 #The user wrap a point, drags it, placing it at P'.
 #The final position of the point is P' but the user cannot modify the first term P(B) so the second term (b) becomes: P(animated)=P'-P(B).

@@ Line 42: / Line 42: @@
 === How does a Vertex move under the influence of a single Bone? ===
-Conceptually a bone has influence over a point doing [http://en.wikipedia.org/wiki/Affine_transformation affine transformation] of it position according to the bone values. See also [http://en.wikipedia.org/wiki/Transformation_matrix transformation matrix]. A bone produces rotation, scale and translation to a point in this way:
+Conceptually a bone has influence over a point by doing [http://en.wikipedia.org/wiki/Affine_transformation affine transformations] of its position according to the bone values. See also [http://en.wikipedia.org/wiki/Transformation_matrix transformation matrix]. A bone produces rotation, scale and translation to a point in this way:
 [[Image:Bonesimulation2.png]]
-As you can see the values that produces the movement of the point P to the point P' are given by:
+As you can see the values that produce the movement of the point P to the point P' are given by:
 *Translation of the bone: O'-O
@@ Line 52: / Line 52: @@
 *Scale of the bone: s=length'/length
-Then, given a Point P in the 2D space and known the translation, rotation and scale of the bone you can calculate the new point position P' using this formulation:
+Then, given a Point P in the 2D space and knowing the translation, rotation and scale of the bone you can calculate the new position P' using this formula:
 <math>
@@ Line 68: / Line 68: @@
 & 0 & 1
 \end{bmatrix}
-  </math> Rotate an angle of amount '<math>-\alpha</math>' around the origin. That aligns the bone with the X axis
+  </math> Rotate by an angle of amount '<math>-\alpha</math>' about the origin. That aligns the bone with the X axis
 <math>
@@ Line 76: / Line 76: @@
 & 0 & 1
 \end{bmatrix}
-  </math> Scale a value of 's=length'/length' in the direction of X (the current direction of the bone).
+  </math> Scale by a value of 's=length'/length' in the direction of X (the current direction of the bone).
 <math>
@@ Line 84: / Line 84: @@
 & 0 & 1
 \end{bmatrix}
-  </math> Rotate an angle of amount '<math>\alpha'</math>' around the origin.
+  </math> Rotate by an angle of amount '<math>\alpha'</math>' about the origin.
 <math>
@@ Line 94: / Line 94: @@
   </math> Translation to new position O'
-Then the transformation from P to P' is give by the matrix multiplication:
+Then the transformation from P to P' is given by the matrix multiplication:
 <math> P'=P\cdot T(-O)\cdot R(-\alpha)\cdot S_x(length'/length)\cdot R(\alpha^')\cdot T(O^')</math>

@@ Line 147: / Line 147: @@
 Also, the final position P' of a point that moves influenced by a single bone is given by the original point position relative to the bone in its "original" position (P).
-What I'm trying to explain you is that it is needed that the setup of the bone (position, alpha and length) and the setup position of the point (P) to be moved would be given to calculate the final position of the point (P') given a moved position of the bone (position', alpha', length')
+What I'm trying to explain is that it is needed that the setup of the bone (position, alpha and length) and the setup position of the point (P) to be moved would be given to calculate the final position of the point (P') given a moved position of the bone (position', alpha', length')
 Said this and considered the information of the previous sections, the bones and points that are influenced from the bones should have the following parameters: