A gouge is largely specified by its sweep and width, the sweep being a manufacturers number for a certain configuration of a gouge. The width is measured between the outside tips of a gouge's cutting edge.
Sweeps #1 - #10 usually describe straight gouge irons with a curvature to the cutting edge from straight (#1) to nearly half-circle (#10). Other sweeps are mostly up to the manufacturer and differ and even between manufacturers, a #3 is most likely not the same.
Gouge cutting edges commonly have the bevel on the outside, but there are in-cannel gouges that are sharpened on the inside used e.g. for flutes.
There are straight irons, gently bend ones, heavily bent ones (also called spoon bent or swan neck) and back bent ones (which are basically spoon bent/in-cannel gouges)



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The U-shaped gouge is also called a veiner.

V-tools come in different angles.
Get carving gouges by Pfeil, Stubai, MStein, Ashles Iles, Two Cherries.
see chip-carving.
Mora 120 (shorter) and 106 (longer) are good beginner knives. There's also 122 (straight edge) and 105 (guard).
The catalogs are great places to look for sweep numbers and profiles.
Here's an interesting take on the Sweep system, apparently called the "Sheffield List":
| Sweep | height in percent of #9 | |
|---|---|---|
| #1 | 0% | |
| #2 | 1-4% ? | |
| #3 | 5% | |
| #4 | 10% | |
| #5 | 15% | |
| #6 | 25% | |
| #7 | 50% | |
| #8 | 75% | |
| #9 | 100% | Standard, basically a semi circle |
| #10 | 110-112% | |
| #11 | 125% |
(archived source: https://web.archive.org/web/20220404054701/https://www.altebeitel.de/index.php/bauformen/holzbildhauer)
Visualisierung:

Interactive tool to visualize and calculate sweeps using the width and percentage of a half circle (or an ellipse with a given aspect ratio):
Checking this concept against the published Pfeil (Swiss Made) gouge sweep tables from the catalog, it works fairly well for sweeps 1-9. Sweep 11 is different as it does not follow a circle, but an ellipse: it matches one with a ratio of 1:2 (3:7 might be more accurate) fairly well with the segment being around 40% (30% at 3:7) of that ellipse.
Another interesting approach, e.g. https://klensmide.se/produktfiler/sticktabell-hk-nedbockat-jarn.pdf
radius/width, e.g. 35/30 is the portion of a 35mm circle, that's 30mm wide. This results in the curve heavily depending on the width, which means sweeps are barely comparable; it's a very accurate specification, though.