Why won't Curve Parameter scale my instances?

Question

I am trying to make (simplified example) the following:

• A curve
• with icospheres along its length
• where the radius and distance between icospheres is proportional to the distance from the start-point of the curve

What I'd like:

I understand that the Curve Parameter node is responsible for telling me how far along a curve I am, on the interval [0-1]. So my intuition is to pipe the curve param into both the scale of the instances, as well as the length offset of the Resample Curve node:

!IMPORTANT! If you emulate my tree, make sure you have an Add node between Curve Parameter and Resample, or you'll get a huge number of icospheres and things will get slow :)

But this does not appear to work: Not only is the distance offset between spheres the same; they also are not scaled at all! (I've already confirmed that this is not due to imperceptibly small differences in scale)

Am I misunderstanding the Curve Parameter node? Is there another way I could better accomplish this?

Answer 1

This solution is 100% accurate only for a straight line. Dealing with bezier lengths is very tricky and requires recursion, so really for a 100% accurate result on a curve I would expect only a Python solution to work...

Let's say each next icosphere is twice bigger. Let's manually calculate icos. positions.

If you have a single icosphere, you want to position it in the middle of the curve, 50% of it's length, or 1/2.

If you have two icos., you want the first to take 1/3, and the other to take 2/3 of the curve. So you want to position them at ${1\over3} \times {1\over2} = {1\over6}$ and ${1\over3} + {2\over3} \times {1\over2} = {2\over3}$. Multiplying by half, because the center is in half of the taken space.

3 icospheres positions:

• ${1\over7} \times {1\over2} = {1\over14}$
• ${1\over7} + {2\over7} \times {1\over2} = {2\over7}$
• ${1\over7} + {2\over7} + {4\over7} \times {1\over2} = {3\over7} + {2\over7} = {5\over7}$

4 icospheres positions:

• ${1\over15} \times {1\over2} = {1\over30}$
• ${1\over15} + {2\over15} \times {1\over2} = {2\over15}$
• ${1\over15} + {2\over15} + {4\over15} \times {1\over2} = {3\over15} + {2\over15} = {5\over15}$
• ${1\over15} + {2\over15} + {4\over15} + {8\over15} \times {1\over2} = {7\over15} + {4\over15} = {11\over15}$

You probably see a pattern by now:

• Space taken by each instance is a fraction with the numerator being the index raised to the power of 2...

0. $0^2 = 1$
1. $1^2 = 2$
2. $2^2 = 4$
3. $3^2 = 8$

• ...And the denominator being the total number of icos. raised to the power of 2, minus 1.

1. Icos.: $1/1$
2. Icos.: $1/3$, $2/3$
3. Icos.: $1/7$, $2/7$, $4/7$
4. Icos.: $1/15$, $2/15$, $4/15$, $8/15$

• Finally, the offset caused by previous instances follows the same pattern: 0, 1, 3, 7...

All those things can be calculated for each instance independently, without some iteration or recursion that geometry nodes don't yet support. The formula becomes:

$f = {2^{i-1}\over d} + {{2^i\over2}\over d} = {2^{i-1} + {2^i\div2}\over d}$

$d = 2^n -1$

where $f$ is a factor (% of the curve to sample a position at), $n$ is the number of icospheres and $i$ is the index of the icosphere.

Answer 2

The Scale Instances node operates on instances, not geometry, so it cannot be used before the Instance on Points node. There are two ways to scale instances:

Scaling with the Instance on Points node:

This is the simplest way to scale instances. Just plug the scale directly into the Instance on Points node, as shown in this image. Note that the Combine XYZ node is unnecessary, since this is how Blender converts floats to vectors anyway.

Scaling with the Scale Instances node:

Using the Scale Instances node with the curve factor is slightly more difficult. Since the Instance on Points node does not transfer this data to instances, you need to use a Capture Attribute node before the Instance on Points node. Then you can place a Scale Instances node after the Instance on Points node, and use the captured attribute.

Answer 3

You can scale after Instances on Points but if you use a field for the Scale input, you can also scale in Instances on Points. This tree:

produces this output:

if the original curve is an unmodified Bezier circle.

The remaining problem, and it has me stumped, is to scale the distance between the instances. This tree

does scale the length, as you can see by changing the constant in the Multiply node; but it scales the lengths evenly and it should scale by distance of the control point along the original curve.