Using the sketch conic curve function
This is an interesting one and I admit to not fully understanding the result.
The conic curve function in Fusion allows you to set the two end points of a conic curve and then ‘pull back’ the curve to its apex and enter the rho value. When the rho value is entered as 0.5 this will create a parabola. (If you enter rho as >0.5 it will create a hyperbola and if <0.5 it will be an elipse).
With a rho value of 0.5 the focus point is the same distance from the curve vertex as the curve vertex point is to the directrix. The directrix is a mirror line behind the curve that at the parabola vertex is the same distance from the back of the curve as the focus is to the front, hence the 0.5 rho factor. (Best to see the sketches below …..).
If you draw a perpendicular line from the directrix to the curve and then from the curve to the focus, then in a true parabola the length of these two lines should be equal. (This is why parabolic shaped dishes are used for radio communication links. In simple terms, any parallel radio waves incident on the dish will have the same reflected propagation distance to travel to arrive at the focus point where the detector device will be located. There is a cumulative addition of the incident waves at the focus. This is the power gain factor achieved by the dish in amplifing the signal.
All good so far ?
The problem I encountered was I could create a random conic parabolic curve using the Fusion function (two end points and vertex using a rho of 0.5) but the paths from the directrix to the curve and the curve to the focus did not match.
Until quite by chance … I had drawn a parabola conic (0.5 rho) where the diameter of the curve (curve peak to curve peak) was twice the vertex peak to the directrix line. Using this ratio always gave me equal path lengths between focus and curve and the curve to directrix.
Conclusion – the conic curve function in Fusion is not necessarily producing the result you might expect of a true parabola shape. I think it is almost certainly to do with the equation of a parabolic curve which the Fusion function is maybe not fully addressing.
Update : Looking back to my amateur radio microwave activity I remember the true focus of a dish was derived from the formula f=D2/16d where D is the dish diameter and d is the depth of the dish. From this you can calculate the focal ratio f/D. For efficient illumination of a centre fed dish the f/D ratio had to lie between 0.3 and 0.45.
By chance my choice of 120mm diameter and 30mm dish depth provides a 30mm focal length and the Fusion conic curve graphing is correct and gives an f/D 0.25. I think for any other f/D ratio the Fusion conic curve will not be an exact match to a true parabola curve (that is the focus to curve and curve to directrix path lengths will not be equal). I think this is because Fusion assumes the focus point is on the same axis as the end points of the conic curve. On this basis any conic curve created in Fusion with an expected rho value of 0.5 (a parabola) must have the directrix positioned at a distance equal to half the distance between the two end points. The conic curve is then ‘pulled back’ to the directrix line and the rho value of 0.5 entered. This could be automated using parametric functions. If the focus is not on the same axis as the curve end points some head scratching will be necessary and any resulting conic curve will not be a true parabola.
A construction related comment.
Never trust the accuracy of a floating Point placed based on just a selected grid position in Fusion. You must dimension lock the Point back to the sketch reference otherwise any resulting sketch using the Point will not be constrained. The parabola sketch is an example. I placed the two end Points for the curve, the curve centre and the directrix centre as my initial sketch postions. Then using the conic curve function clicked on the two curve ends and pulled the vertex back to the directrix point and entered the rho value of 0.5. The curve line immediately came up in black to indicate it was fully constrained. If any of the Points had not been dimension locked, the curve will appear blue – unconstrained.
To conclude this waffle and complete a solid version of the parabola, use the offset command to draw a parallel line behind the parabola curve and then ‘seal’ the end of the two curves with a short line and also add a short centre dividing line. You can now select half of the solid in extrude mode and rotate around the centre line axis to give a complete solid parabola. Here is a step by step process.
Here is the resulting solid parabola. This might be an interesting 3D print …..
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