The curvature tool in Illustrator is found in the toolbar near the top of the panel.
Select, there are no options available for this via the icon so you can't double click it to display some options etc. Set the color to nil for the fill (you can have a color fill as well but unless you create a closed path, the fill may be of less use. Set the stroke to black and 7pt say for the width. Click on the art board and then click again and after that you will see a preview blue line showing the direction of the curve which can be modified before you click to add the third point to the curve. To escape from adding more points to the curve, press the escape key and it will stop the blue preview path.
You will see the line has a number of points along the path. You can select any of those points and shift them, you can also shift them along the path as well.
You will notice the blue preview keeps re-appearing, you can remove this by going to the preferences and selection category and the bottom part of the panel and 'rubber band' , turn it off (I must admit, it is useful but does get in the way as well)
You can if you wish, close the path by clicking the first point and that case the fill is more useful.
You can double click a point and turn it into a corner sharp point, you can double click it again to turn it into a smooth curve again. You can click anywhere else on the curve (not on an existing point) and a new point to the path will be added and it will slightly change the shape You can also click a point and then press the delete key to delete it. You can also use the direct selection and move the points as well as modify the direction handles. You can also then change the corner value by using the interactive control as well as changing the corner setting in the top of the control bar. You can also use the direct selection to drag the path between the points and manipulate that in many ways.
Make certain you have deselected the curve before starting a new one because it will just try and connect it to the existing one
You can also use it on existing paths such as star design or ellipse created using the ellipse or arc designs and more. You can manipulate the points as well add additional points and shift the curve using it.
You can create the curve using it, the stroke can be further manipulated by the width feature or by adding width profiles and the line can be changed in countless ways. You can, once you are happy with your artwork expand this by using the expand appearance found in the object menu though in most cases it generates a lot of points.
You can also select brushes via the brushes panel and add these to the artwork and you can still continue to add additional points as the line is still live and can be edited over and over until you expand it.
You can add effects via the effect menu which can all be edited via the appearance panel found in the window menu. Go to the effect menu and distort and transform and then transform and with the path selected, set the copies and the move value as required to create multiple copies of the same curvature design stroke
Goto the appearance panel with the stroke and then right side menu and new stroke. Set the color for the path. Goto the effect menu and select path and offset path. Set to a value in the offset that displays both the original as well as the duplicate. You will now see the offset in the appearance panel and it can be double clicked and edited at any time. You can also continue to add and remove and move points.
Select it and create a path, set the color, width profile etc. Go to the object menu and repeat and radial and set the radius and instances. Double click one of the instances and you can continue to edit the path. You can then exit the repeat isolation mode via the top left arrows. Once you have exited out of the repeat object then you can no longer edit the path but if you want to extend it, you can again double click on one of the instances and change the artwork. All the instances will be updated.
The design can be expanded via the object menu and expand command. Select it and set the stroke color etc and click the art board and continue to draw, make a tangle of points by adding points back and forth across the art board. You can then use the live paint bucket to fill the cells of the image. You can warp the design using live effects found via the effect menu. You can also warp the design using warping features such as the wrinkle and crystallize and scallop.