Z Rotation Cubing
2021年11月12日Register here: http://gg.gg/wuh1j
*Z Rotation Cubing
*Z Rotation Cubing
I made all the different corners (I calculated them by getting the center X/Y/Z of the cube, and the radius, and adding or subtracting the radius to the center X/Y/Z based on what corner of the cube I wanted), and I followed the lesson to the best of my ability (it is in JavaScript, while I am using C# /w SFML). The good news is that learning onlyif six intuitive letters is sufficient to solve the cube but if you are a speedcuber you should learn the advanced notation which you can access from this page. A single letter by itself refers to a clockwise face rotation in 90 degrees (quarter turn): F R U L B D.
Rotating things in three dimensions sounds complicated and it can be, but there are some simple rotations. For example, if we imagine rotating our cube around the z-axis (which points out of the screen), we are actually just rotating a square in two dimensions.There is a reason to learn trigonometry
We can simplify things further, by just looking at a single node at position (x, 0). Using simple trigonometry we can find that the position of the point after rotating it by θ around the origin is (x’, y’), where, y’ = x × sin(θ)
If you don’t understand where these equations came from, this video might help.Rotating a point about the origin
The example above allows us to rotate a point that starts on the x-axis about the origin, but what if it isn’t on the x-axis? This requires some slightly more advanced trigonometry. If we call the distance between the point (x, y) and the origin r, and the angle between the line to (x, y) and x-axis, α then,Z Rotation Cubingx = r × cos(α)
If we rotate by β to point (x’, y’), then,x’ = r × cos(α + β)
Mit einzahlung. Using the trigonometric addition equations (derived here and here), we get,x’ = r × cos(α) cos(β) - r × sin(α) sin(β)
Substituting in the values for x and y above, we get an equation for the new coordinates as a function of the old coordinates and the angle of rotation:x’ = x × cos(β) - y × sin(β) Writing a rotate function
Now we know the mathematics, we can write a function to rotate a node, or even better, our array of nodes, around the z-axis.Our cube is slightly more interesting, but not much.
This function loops through all the nodes in the node array, finds its current x and y coordinates and then updates them. We find sin(theta) and cos(theta) outside the loop so we only need to calculate it once.
We can test the function by calling it. Try rotating the cube by 30 degrees.
You can see the complete code here.
See? Trigonometry can be useful!Rotating in three dimensions
We can now rotate our cube in two dimensions, but it still looks like a square. What if we want to rotate our cube around the y-axis (veritcal axis). If we imagine looking down on our cube as we rotate it around the y-axis, what we would see is a rotating square, just like we do when we rotate about the z-axis.
So if we imagine our relabelling our axes, so the z-axis becomes the y-axis, we can come up with a new function for rotating around the y-axis. In this case, the y-coordinates of the node do not change.
And we can use the same argument to create a function that rotates our cube around the x-axis:
Again, we can test the code by calling the function.
You can see the complete code here. Try using the slider to change the values in the function calls.User interaction
We can rotate the cube by adding function calls, but it’s a lot more useful (and satisfying) if we can rotate it using the mouse. For this we need to create a mouseDragged() function. This function is automatically called whenever the mouse is dragged.
mouseX and mouseY are built-in variables that contain the current position of the mouse. pmouseX and pmouseY are built-in variables that contain the position of the mouse in the previous frame. So if the x-coordinate has increased (we move the mouse right), we send a postive value to rotateY3D() and rotate the cube counter-clockwise around the y-axis.
You can see for yourself here.
The rotate3d()CSSfunction defines a transformation that rotates an element around a fixed axis in 3D space, without deforming it. Its result is a <transform-function> data type.
Occasion helsinki. The source for this interactive example is stored in a GitHub repository. If you’d like to contribute to the interactive examples project, please clone https://github.com/mdn/interactive-examples and send us a pull request.
In 3D space, rotations have three degrees of liberty, which together describe a single axis of rotation. The axis of rotation is defined by an [x, y, z] vector and pass by the origin (as defined by the transform-origin property). If, as specified, the vector is not normalized (i.e., if the sum of the square of its three coordinates is not 1), the user agent will normalize it internally. A non-normalizable vector, such as the null vector, [0, 0, 0], will cause the rotation to be ignored, but without invaliding the whole CSS property.Note: Unlike rotations in the 2D plane, the composition of 3D rotations is usually not commutative. In other words, the order in which the rotations are applied impacts the result.SyntaxZ Rotation Cubing
The amount of rotation created by rotate3d() is specified by three <number>s and one <angle>. The <number>s represent the x-, y-, and z-coordinates of the vector denoting the axis of rotation. The <angle> represents the angle of rotation; if positive, the movement will be clockwise; if negative, it will be counter-clockwise.ValuesxIs a <number> describing the x-coordinate of the vector denoting the axis of rotation which could between 0 and 1.yIs a <number> describing the y-coordinate of the vector denoting the axis of rotation which could between 0 and 1.zIs a <number> describing the z-coordinate of the vector denoting the axis of rotation which could between 0 and 1.aIs an <angle> representing the angle of the rotation. A positive angle denotes a clockwise rotation, a negative angle a counter-clockwise one.Cartesian coordinates on ℝ2This transformation applies to the 3D space and can’t be represented on the plane.Homogeneous coordinates on ℝℙ2Cartesian coordinates on ℝ31+(1-cos(a))(x2-1)z·sin(a)+xy(1-cos(a))-y·sin(a)+xz·(1-cos(a))-z·sin(a)+xy·(1-cos(a))1+(1-cos(a))(y2-1)x·sin(a)+yz·(1-cos(a))ysin(a)+xz(1-cos(a))-xsin(a)+yz(1-cos(a))1+(1-cos(a))(z2-1)t0001Homogeneous coordinates on ℝℙ3ExamplesRotating on the y-axisHTMLCSSResultRotating on a custom axisSpecificationsSpecificationStatusCommentCSS Transforms Level 2
The definition of ’rotate3d()’ in that specification.Editor’s DraftInitial definitionBrowser compatibility
BCD tables only load in the browserSee also
Register here: http://gg.gg/wuh1j
https://diarynote.indered.space
*Z Rotation Cubing
*Z Rotation Cubing
I made all the different corners (I calculated them by getting the center X/Y/Z of the cube, and the radius, and adding or subtracting the radius to the center X/Y/Z based on what corner of the cube I wanted), and I followed the lesson to the best of my ability (it is in JavaScript, while I am using C# /w SFML). The good news is that learning onlyif six intuitive letters is sufficient to solve the cube but if you are a speedcuber you should learn the advanced notation which you can access from this page. A single letter by itself refers to a clockwise face rotation in 90 degrees (quarter turn): F R U L B D.
Rotating things in three dimensions sounds complicated and it can be, but there are some simple rotations. For example, if we imagine rotating our cube around the z-axis (which points out of the screen), we are actually just rotating a square in two dimensions.There is a reason to learn trigonometry
We can simplify things further, by just looking at a single node at position (x, 0). Using simple trigonometry we can find that the position of the point after rotating it by θ around the origin is (x’, y’), where, y’ = x × sin(θ)
If you don’t understand where these equations came from, this video might help.Rotating a point about the origin
The example above allows us to rotate a point that starts on the x-axis about the origin, but what if it isn’t on the x-axis? This requires some slightly more advanced trigonometry. If we call the distance between the point (x, y) and the origin r, and the angle between the line to (x, y) and x-axis, α then,Z Rotation Cubingx = r × cos(α)
If we rotate by β to point (x’, y’), then,x’ = r × cos(α + β)
Mit einzahlung. Using the trigonometric addition equations (derived here and here), we get,x’ = r × cos(α) cos(β) - r × sin(α) sin(β)
Substituting in the values for x and y above, we get an equation for the new coordinates as a function of the old coordinates and the angle of rotation:x’ = x × cos(β) - y × sin(β) Writing a rotate function
Now we know the mathematics, we can write a function to rotate a node, or even better, our array of nodes, around the z-axis.Our cube is slightly more interesting, but not much.
This function loops through all the nodes in the node array, finds its current x and y coordinates and then updates them. We find sin(theta) and cos(theta) outside the loop so we only need to calculate it once.
We can test the function by calling it. Try rotating the cube by 30 degrees.
You can see the complete code here.
See? Trigonometry can be useful!Rotating in three dimensions
We can now rotate our cube in two dimensions, but it still looks like a square. What if we want to rotate our cube around the y-axis (veritcal axis). If we imagine looking down on our cube as we rotate it around the y-axis, what we would see is a rotating square, just like we do when we rotate about the z-axis.
So if we imagine our relabelling our axes, so the z-axis becomes the y-axis, we can come up with a new function for rotating around the y-axis. In this case, the y-coordinates of the node do not change.
And we can use the same argument to create a function that rotates our cube around the x-axis:
Again, we can test the code by calling the function.
You can see the complete code here. Try using the slider to change the values in the function calls.User interaction
We can rotate the cube by adding function calls, but it’s a lot more useful (and satisfying) if we can rotate it using the mouse. For this we need to create a mouseDragged() function. This function is automatically called whenever the mouse is dragged.
mouseX and mouseY are built-in variables that contain the current position of the mouse. pmouseX and pmouseY are built-in variables that contain the position of the mouse in the previous frame. So if the x-coordinate has increased (we move the mouse right), we send a postive value to rotateY3D() and rotate the cube counter-clockwise around the y-axis.
You can see for yourself here.
The rotate3d()CSSfunction defines a transformation that rotates an element around a fixed axis in 3D space, without deforming it. Its result is a <transform-function> data type.
Occasion helsinki. The source for this interactive example is stored in a GitHub repository. If you’d like to contribute to the interactive examples project, please clone https://github.com/mdn/interactive-examples and send us a pull request.
In 3D space, rotations have three degrees of liberty, which together describe a single axis of rotation. The axis of rotation is defined by an [x, y, z] vector and pass by the origin (as defined by the transform-origin property). If, as specified, the vector is not normalized (i.e., if the sum of the square of its three coordinates is not 1), the user agent will normalize it internally. A non-normalizable vector, such as the null vector, [0, 0, 0], will cause the rotation to be ignored, but without invaliding the whole CSS property.Note: Unlike rotations in the 2D plane, the composition of 3D rotations is usually not commutative. In other words, the order in which the rotations are applied impacts the result.SyntaxZ Rotation Cubing
The amount of rotation created by rotate3d() is specified by three <number>s and one <angle>. The <number>s represent the x-, y-, and z-coordinates of the vector denoting the axis of rotation. The <angle> represents the angle of rotation; if positive, the movement will be clockwise; if negative, it will be counter-clockwise.ValuesxIs a <number> describing the x-coordinate of the vector denoting the axis of rotation which could between 0 and 1.yIs a <number> describing the y-coordinate of the vector denoting the axis of rotation which could between 0 and 1.zIs a <number> describing the z-coordinate of the vector denoting the axis of rotation which could between 0 and 1.aIs an <angle> representing the angle of the rotation. A positive angle denotes a clockwise rotation, a negative angle a counter-clockwise one.Cartesian coordinates on ℝ2This transformation applies to the 3D space and can’t be represented on the plane.Homogeneous coordinates on ℝℙ2Cartesian coordinates on ℝ31+(1-cos(a))(x2-1)z·sin(a)+xy(1-cos(a))-y·sin(a)+xz·(1-cos(a))-z·sin(a)+xy·(1-cos(a))1+(1-cos(a))(y2-1)x·sin(a)+yz·(1-cos(a))ysin(a)+xz(1-cos(a))-xsin(a)+yz(1-cos(a))1+(1-cos(a))(z2-1)t0001Homogeneous coordinates on ℝℙ3ExamplesRotating on the y-axisHTMLCSSResultRotating on a custom axisSpecificationsSpecificationStatusCommentCSS Transforms Level 2
The definition of ’rotate3d()’ in that specification.Editor’s DraftInitial definitionBrowser compatibility
BCD tables only load in the browserSee also
Register here: http://gg.gg/wuh1j
https://diarynote.indered.space
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