I’ve been struggling with transformations for a few days now.
And I really needed this post. But it didn’t exist.
So I am writing it now.
If you are having trouble too, this post is for you.
If you have no idea what a matrix is, then I might have to disappoint you.
We are not going to talk about Neo today (although you might want to check out this cool behind the scenes video).
We’re talking about these things:
The matrix above has two rows and three columns. A matrix with three rows and one column (called a column vector or column matrix) is used to describe a point in 3D.
It’s basically an arrow from the origin to this point. It has a direction and a length.
When you multiply a 3×3 matrix with a 3×1 (3 rows, 1 column) vector, the result is another 3×1 vector with a different direction. This is called a transformation.
The length of the vector stays the same when you only change the orientation.
Everything SOLIDWORKS does uses matrices:
If you already know how matrix multiplication works, you can skip ahead to the next section.
Translating a point is pretty simple to do. Just add two column vectors to get the sum.
You can only sum matrices of the same size. Each element is the sum of the element in the parts, for example 1+4 = 5.
You can visualize summing vectors by stacking two arrows. The sum is the endpoint of the top arrow.
(The image below does not use the same values as the example above by the way)
I can best start explaining rotation matrices in 2D.
To rotate a x,y vector with angle theta, you multiply the vector with a rotation matrix:
Wolfram Alpha is nice enough to also show how you can manually calculate the X and Y values of the rotated point.
To calculate the X value (the first row), multiply each element in the first row of the matrix by the first column of the original point. Then you sum the results.
To calculate Y (the second row), multiply each element in the second row of the matrix with the first row of the vector. Then sum them again.
I also found this example on StackExchange, which nicely visualizes rotating a square:
Rotations around an axis in 3D use the same rotation matrix as in 2D, but with an added row and column.
When you rotate something around the X-axis, the X-value remains the same. That’s why the first entry is one and all other values in that row and column are zero.
You see the same pattern for rotations around Y and Z. Sometimes the sign for the sine is flipped though:
Image source: Wikipedia
I have been coming back to this post for help regularly. Yesterday I noticed my 90-degree rotation around Z gave negative values for the sin(theta) values.
It turns out that there is a difference between the matrix to transform a point from X to X’ is different than the matrix to transform the coordinate system R3 to R3′.
So there is no need to be alarmed. Just be aware.
Source: Wolfram Mathworld, equation 3-6
To scale a point with regards to the origin, all you need to do is multiply each value by the scale.
Or if you want to use a complete matrix to scale equally in all directions:
The nice thing about 2D and 3D transformations is that you can do the calculations manually on paper as well.
But you can also use the awesome site Wolfram Alpha.
SOLIDWORKS uses 4×4 matrices to define transformations. They call it a MathTransform.
It’s built up out of four sections:
Not used means these values are always zero.
Note that SOLIDWORKS names the elements per submatrix. It always uses this custom order, for example when it returns 16 elements as ArrayData:
Combined, this matrix creates an affine transformation. This basically means that, when you multiply multiple objects with the same transformation, relations between points remains the same and parallel lines and planes remain parallel.
When you do manual calculations, multiply the vector by the rotation first, then scale the result, then translate that result.
To rotate a point (or any other object) around the origin (and keep the scale and the distance to the origin the same), set the values a-i and set the scale to 1:
You can think of each row in these matrices as the direction of an axis. Check out the helpful image that I made below.
The front view uses the standard coordinate system. The top view has the X-axis in the same direction, whereas the right view has the X-axis in Z-direction.
You can also see that the Z-axis in the top view is in the negative Y direction of the front view.
That should make these rotation matrices a little easier to read.
To translate a point in 3D space and keep the orientation and the scale constant, set the rotation matrix to the identity matrix and the scale to 1.
When you multiply a point with this matrix, the point moves j in the x-direction, k in the y-direction and l in the z-direction.
To scale a point position with a factor of two, only set the scale factor.
You need to use the matrix inverse to create a transformation in the other direction.
For example to go from view to model instead of model to view.
These inverse matrices have some nice properties:
Combining a rotation, a translation and a scale is not a problem, just add all of the entries to a single matrix.
But to combine two full matrices, you need to multiply them.
When you multiply two square matrices, the result is a different (composite) square matrix with the same dimensions. You can use MathTransform.Multiply for that within SOLIDWORKS.
According to the SOLIDWORKS support employee I asked for help, the multiplication order does not matter. I still don’t believe this.
One of the first things I learned about matrices is that the multiplication order is of paramount importance. So I decided to test it in SOLIDWORKS.
I created a rotation matrix for a top view (-90 degrees around X) and one for the right view (90 degrees around Y).
Then I multiplied them: top.Multiply(right) and right.Multiply(top). The resulting matrix is not the same.
Rotation for top view * rotation for right view
I have also tested this within SOLIDWORKS with a point. Turns out the order is important after all.
So be careful with this.
The multiplication order SOLIDWORKS uses is from left to right, just as you would read it:
top.Multiply(right) equals doing [top]*[right].
The SOLIDWORKS help does not state this anywhere though.
Here is a code example on how to create a MathTransform. I prefer to write in C#, but this is in VBA because that is still used by many.
Dim swApp As SldWorks.SldWorks
Dim mathUtility As mathUtility
Dim transform As MathTransform
Set swApp = Application.SldWorks
Set mathUtility = swApp.GetMathUtility
Dim xValue As Double, yValue As Double, zValue As Double
Dim transformArray(15) As Double
'Rotation. Note that there are three assignment per line to shorten the macro
transformArray(0) = 1: transformArray(1) = 0: transformArray(2) = 0:
transformArray(3) = 0: transformArray(4) = 1: transformArray(5) = 0:
transformArray(6) = 0: transformArray(7) = 0: transformArray(8) = 1:
transformArray(9) = xValue: transformArray(10) = yValue: transformArray(11) = zValue:
transformArray(12) = 1
Set transform = mathUtility.CreateTransform(transformArray)
Dim transformData As Variant
transformData = transform.ArrayData
This code gets the MathUtility, creates an array that contains the data, then creates a MathTransform from that array.
You can verify the data using mathTransform.ArrayData, which returns a double array with 16 numbers as a variant.
When you do something in the assembly or the part with regard to the origin, this is called Model space.
A definition of a space would be a set of points, with a relation between those points. It basically says how the origin and the coordinate system is created in relation to another coordinate system.
Here is a list of spaces that are used in SOLIDWORKS. The names are not officially defined anywhere though.
I’m trying to list all spaces that are used within SOLIDWORKS, so if you miss one, please get in touch and I’ll add it.
Also contact us if I made a mistake anywhere in this post, which is pretty likely to happen…
I’m trying to make this an exhaustive list.
So if you’re missing a useful property or method here, get in touch and I’ll add it.
I recently needed this matrix when converting view points in a drawing to model space (this matrix combined with ModelToSheetTransform actually).
It only contains an identity matrix for the rotation, plus a value for the current scale. It got pointed out to me by Artem Taturevych on the SOLIDWORKS forums and it’s used in one of his examples as well.
So if you’re stuck and you only need to scale your object, this matrix might be the one you need.
When you get the position of a point in a sketch, it’s pretty obvious that this point is in sketch space.
To get back to model space, multiply the position with the inverse of ModeToSketchTransform. It’s that simple usually.
But there are more complex cases. I’ll be adding to this list when I find new situations or when you send me one:
The more I work with the SOLIDWORKS API, the more I run into transformations and their matrices.
While developing our drawing automation tool Drew, I needed to scale, rotate and translate blocks. Right now I’m converting views and mouse coordinates to sheet space.
I also created software for a client that creates thousands of motion studies. There I rotated components, ran motion studies, then analyzed and compared the resulting orientations again. This was a major (and fun) exercise in transformation matrices.
Yet I keep fighting with transformations. So I will probably read this post to solve my own problems in the near future.
I called this post “A complete overview” and I mean it. But it’s not complete yet.
So if you have a strange use case or a question that you would like to add to this post, email me at email@example.com so I can add it.