I want to move an anchor point along its current path - I mean moving anchor points keeping their current angle. Look at this picture:
According to above picture, let's say we want to move anchor point 1, but as you can see there is no option (or I cant find the option) to do this while preserving the angle - with the shift key, the anchor point snaps to 45° angles...
Illustrator does not have a easy shortcut to do this. However, you can do this with the scale tool. While this sounds as a weird thing, why it works becomes apparent once you realize we move the pivot point to previous control point. Here is what you do:
(Make sure smart guides is on)
Select the point you want to move.
Activate scale tool (S)
Move the anchor point to the position of prevous point
Drag aloing the direction, a indication uniform should be present and snap while you are dragging in right driection.
There is also another way to do this with Smart Guides.
If you add a couple of anchors somewhat down a bit from the curve anchors (added anchors circled in red below), and convert those to corner points, then you can use the smart guides and it will show a Line Extension guide as you click and drag.
There is a (paid) plug-In from ASTUTE for this. It's called VectorScribe and can do much much more. The function you are searching for calls trim and extend Paths. With this plug-in you may also move Anchor points without changing the Path.
You can use the free trial first or wait for special sales / discount.