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In Fusion 360, I tried some add-ons, like the "Equation driven curve", but the distance between each revolution is a constant. To achieve a constant-angle-of-attack spiral, the revolution distance shouldn't be a constant value. Is it manageable in this way? I also tried it in Blender, but after 3 turns, its angle of attack changes a lot. How would it be possible to design a spiral like that? Should I try it in AutoCAD?

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  • Wrong place to ask, graphic design is about visual communication not cad design. engibeering, blender or marhematics
    – joojaa
    Feb 5, 2022 at 12:25
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    There's a Blender Stack Exhcange. Ask there instead. CAD software is off-topic too.
    – Billy Kerr
    Feb 5, 2022 at 13:50

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You used term angle of attack which sounds like it's taken from turbine or airplane propeller engineering. Some of us may well know something of it, but most of us in GDSE draw and mangle 2D images and make layouts, so do not expect a flood of answers.

I am not an exception so you get only a guess. The guess is: You want a distorted tapered helix which is still on a cone, but it's not linear. The pitch changes non-uniformly so that a short piece of curve whips air by being positioned to a certain predefined angle against the airflow when the spiral is rotated around its axis:

enter image description here

Assuming that guess is OK and using the Z-axis of the ordinary XYZ coordinate system as the spiral axis we can write a few facts

  1. The curve is on a cone. It means that the radius of the spiral (=R) changes linearly along the Z-axis. We can write for example R = AZ where A is a constant. If A=0,5 the spiral radius grows 0,5 mm for every millimeter rise along the Z axis.

  2. The XY-plane projection of the spiral must be logarithmic spiral.

Why? Let's assume the previous green piece of the spiral was got by advancing along the spiral a short distance which changes the placement in the XY plane polar coordinates a small angle dV (in radians). The length of the XY-plane projection of the green curve is RdV:

enter image description here

But in the guess of the meaning of the angle of attack the rotation also happened in XY-plane i.e. around the Z-axis. The tangent of the angle of attack is the same as the rise rate of the spiral, =k in the previous image. We have differential equation (dZ/dV)=kR (see NOTE1)

We can combine it with the"sitting on a cone" -condition R=AZ and get a new differential equation (dZ/dV)= kAZ. From elementary differential equation solution tables one can see, that Z coordinate of the curve must depend on the place in XY plane as follows: Z=(Zo)exp(kAV). Zo is a freely selectable constant. It's the starting elevation of the curve.

So, the full (guessed only, sorry!) set of the equations is:

Z = (Zo)exp(kAV)

Y = A(Zo)exp(kAV)sin(V)

X = A(Zo)exp(kAV)cos(V)

Zo = the elevation of the starting point. There V=0.

k = the tangent of the angle of attack; A = the radius increasing rate along the Z-axis.

It starts to expand from radius A(Zo) upwards and outwards if the parameters are positive.

If one knows the equations for an ordinary Archimed spiral he obviously spots immediately the exponential growth of the radius which makes the spiral logarithmic in the XY-plane. If he doesn't, my whole story probably looks pure nonsense for him.

Let's draw a sample, say 10 turns of it - that's about V = from 0 to 62,8 radians, it's 2pi radiand per a turn.

Let's have radius growth rate A= 0,5 mm per a millimeter, starting elevation Zo = 10 mm (or equivalently starting radius is 5 mm) and the tangent of angle of attack k = 0,1 (angle of attack is about 6 degrees).

enter image description here

It's drawn in DesignSpark Mechanical (freeware), which has recently got a possibility to input curves as equations. I guess that Blender and serious CAD programs allow the same, but with a richer set of available math expressions and scripting. Fortunately this didn't need anything complex.

A view straight against the XY plane shows it could be a logarithmic spiral which is projected on a cone. That's another way to draw it if you have a program which projects curves on surfaces and lets one to extract the result.

enter image description here

Logarithmic spiral is impossible if one wants it start from radius =0. The spiral would be infinitely dense at the tip and that's not presentable as numbers.

NOTE1: This is far from rigorous math. But the pioneers of calculus worked approximately like this i.e. used infinitesimal pieces of curves as lines to be able to apply simple geometric and trigonometric facts to write differential equations for things. The method contains hidden logical contradictions, but it's still used in practical applications of calculus- for ex. in teaching physics.

A rough version of the spiral can also be drawn in Illustrator as a 2D image. It's based on 3D effect "Revolve", so one can decide the watching direction and perspective easily. An example:

enter image description here

The spiral can be got by mapping a set of lines as "art" onto a revolved cone. The mapping dialog is shown in the middle. In the right the revolved object is expanded and all grey parts are deleted.

To make the grey parts deletable one must change the shading mode to "No shading", Apply Object > Expand Appearance, remove the clipping mask and Ungroup a few times.

Expanding the appearance unfortunately splits also the curve to numerous pieces which actually are no more curves, but filled areas. That's why the mapped art line set was made extremely thin (=0,1 pt) and 1pt stroke is inserted after the expanding and removing all grey parts. The final result is in the right.

The mapped art line set was made as follows:

enter image description here

  1. Start by drawing a slightly tilted line

  2. Make progressively more tilted copies of the line which are horizontally as long. Object > Transform > Scale > Horizontally 100%, Vertically 140% > Copy makes one copy. Pressing repeatedly Ctrl+D makes more of them.

  3. Move the lines so that the left end of the next more tilted line is exactly at the same y-coordinate as where'r the right end of the previous line. The horizontal line segments are only for helping the exact snapping with Smart Guides and Snap to Point =ON. Align all lines to the left.

  4. The red lines are deleted and the black line set has got a new thin (=0,1pt) stroke. The line set is dragged to the Symbols collection and used as mapped art in the 3D revolve effect.

The revolved shape is a strokeless rectangular triangle colored to light grey.

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