The kink you can see but cannot find

Three grades of smooth

Two curve segments meeting at an anchor can agree to different degrees, and the words for it come from geometry rather than from software menus.

• G0: the segments meet at the same point. That is all. A corner.
• G1: they meet and their tangent directions are the same. The handles are collinear: a straight line through the anchor. No corner, but the curvature can jump.
• G2: they meet, the tangents agree, and the curvature agrees. The curve bends by the same amount on either side.

Vector editors give you G0 (cusp) and G1 (smooth). None of the common ones enforce G2. Illustrator, Inkscape, CorelDRAW and Affinity will all happily let you build a G1 join where the curvature jumps from tight to loose, and your eye will catch it as a flat spot or a crease even though no corner exists.

How to see it

The trick that industrial designers use and graphic designers mostly do not: add a thick stroke and look at the inside edge. Where curvature jumps, the inner offset of the stroke develops a visible corner even though the centreline is smooth. Put a 6 mm stroke on your curve, zoom in, and the crease announces itself.

A second test: mirror it. Duplicate the shape, flip it horizontally, butt the two together. Any asymmetry or curvature mismatch at the join becomes obvious because now there are two of them.

The common causes, in order of frequency

Unequal handle lengths across a smooth node. The handles are collinear so the join is G1, but one is 5 mm and the other is 40 mm. The curve arrives bending hard and leaves bending gently. Fix by matching the lengths, or by making the node symmetric rather than merely smooth: Inkscape’s Shift+Y makes handles symmetric, Shift+S makes them merely collinear.

A straight segment meeting a curve. A line has zero curvature. Any curve meeting it has non-zero curvature. The jump is mathematically unavoidable at the anchor, so what you control is how gentle it is: make the curve’s handle leaving that anchor long, so it starts nearly straight. A short handle there produces the tell-tale “the arc starts too suddenly” look you see on cheap rounded rectangles.

A node that is not quite at the extreme. If the topmost point of a curve is 0.3 mm to the left of your anchor, the handle cannot be horizontal and still be correct, and every attempt to smooth it fails. Move the anchor to the extreme first, then straighten the handle.

Where it shows up in real work

• Rounded rectangles. A plain rounded rectangle is four quarter-circles joined to four straight lines: eight G1 joins with a curvature jump at each. At large sizes on a sign, you can see all eight. This is precisely why Apple’s superellipse corners and CSS’s “squircle” exist, and why a serious logo with rounded corners is drawn with manually tuned handles rather than a corner-radius field.
• Letterforms. The join between a stem and a bowl is where a badly digitised typeface betrays itself. Set a large o, b and S and look at the inside of the curve.
• Vehicle graphics and signage. A swoosh 3 m long makes a 1% curvature mismatch a 30 mm visible flat.

The practical fix list

1. Put anchors only at extremes and at genuine corners.
2. Make smooth nodes symmetric unless you have a reason not to.
3. Where a curve meets a straight, lengthen the handle on the curve side.
4. Stroke it thick and look at the inside edge.
5. If it still creases, remove the anchor entirely and let one longer segment do the work. Fewer anchors means fewer joins means fewer places for curvature to jump.

What the free tools can and cannot do

Inkscape’s node tool has explicit make symmetric and make smooth commands and a Ctrl+L simplify. It has no curvature comb and no G2 tools. Illustrator has neither either. Genuine curvature analysis lives in CAD: FreeCAD and Blender both show curvature, and Blender’s curve editor will display a curvature comb on a Bézier.

For logo work you do not need CAD. You need the thick-stroke test and the discipline to use three anchors where you wanted six.