A convex edge of equal edge angle to a "V" or linear edge is, in fact, thinner.
There are a couple of important reasons why this is
inaccurate.
1) the image you present is a false representation. The left image suggests that material can be added to the edge by abrasion, a physically impossible absurdity that needn't even be mentioned. The image should instead show the reality in which the apex is brought back. The right image is ALSO misleading because it implies THREE false things: a) that the bevels are being sharpened at the same angle of incidence (or held angle) - they are NOT; b) that the presented convex and flat grind apex angles are identical - they are NOT; and c) that the geometries of these bevels correspond as similar - they do NOT. The flat bevel which corresponds to the green convex bevel would be drawn from apex to bevel shoulder (the blue dot). Draw that bevel and tell me which is thinner.
I went into this in
another thread, but i will resubmit my explanations here.
Cross-section (A) shows convex- (violet) and flat-ground (gray) bevels with corresponding (i.e. similar) geometries =
equal height and shoulder width. One can imagine each profile being ground from identical billets. Note which profile leaves more metal behind the edge.
Cross-section (B) demonstrates how one would grind a comparable convex bevel (violet) out of an original flat-grind (pink). Again, the gray represents reducing the convex bevel back to flat while matching the geometry (height and thickness) of the original (pink) bevel. The gray and pink profiles are identical, the violet profile is comparable... and it is thicker than the gray profile from apex to shoulder, requires less removal of material from the original pink profile.
Cross-section (C) shows the original convex grind (white, violet-outline), reduction to flat-grind (pink), and further reduction to a thinner convex grind (violet). The violet edge is indeed thinner than the pink
at the pink shoulder.
However, 1) from apex until the orange line denoting tangential separation (~1/2 the height of the pink bevel), the violet convex grind is STILL thicker than the pink flat-grind (pushing the apex of the violet grind to match the pink would obviate this fact), and 2) the pink and violet bevels
do not have similar geometries - the pink is as different from the violet as it is from the gray. As before, the true comparison is between the violet and the gray -
bevels of equal height and shoulder thickness. If the pink and violet shoulders where at the same height, the entire violet blade would be thinner as well (assuming the same primary bevel angle and total blade height)!
The crux of the confusion about supposed "thinner" convex grinds is the angle being measured, or rather NOT measured.
In practice (sharpening, and other practices as well, like aerodynamics), the angle being measured is the "
held angle" or "
angle of incidence" between hone surface (flat gray in the above diagram) and spine-center (red line).
NOTE: If this is NOT the angle you are using to grind your bevel,
then you are very likely not using ANY angle measurement at all but instead merely extrapolating after-the-fact. For example, the violet-line in the diagram is
presumed tangential to the precise apex-angle of the green convex... but the precise apex-angle of the green convex
cannot be measured without precision instruments or precise knowledge of the geometry of the curve(s) at the point of bevel intersection (the true apical angle of incidence of a curved shape). Such measurements are unnecessary for the purpose of this discussion as the measured apex and tangent bevel do not produce a triangle of similar geometry beyond an infinitesimally short shoulder height (i.e. at the point of bevel intersection).
Sharpening angle is measured by width of the blade and distance from spine-center to hone. Draw a chord perpendicular to the hone surface that meets the spine-center line to form a triangle. This triangle is geometrically "similar" to the smaller triangle formed by drawing a chord perpendicular to spine-center that intersects the bevel shoulder (light blue triangle). These triangles are similar because their dimensions are directly proportional, their angles equal - these triangles even share an apex!
Altering the shape of the triangle by increasing or decreasing the height of the bevel along the spine-center WITHOUT a proportional change in shoulder thickness (which necessarily changes the angle of incidence) produces NON-similar triangles. Insistence on correlating non-similar geometric shapes produces this idea of "thinner" convex grinds that contradict geometric and mathematical definitions.
To be clear, the definition of "
convex" is as follows:
curved or rounded outward; (math) a continuous function with the property that a line joining any two points on its graph lies on or above the graph; from Latin convexus = carried out/away from.
"Convex" is defined as away from flat, an alteration of shape that can ONLY be accomplished by an increase in angle, i.e. more obtuse, to a form which lies outside or above the corresponding flat plane. To make a convex bevel thinner than a flat bevel, one MUST change the angle of incidence, but the result is still thicker than the flat bevel ground at that new angle and it is the flat bevel
at that angle which informs the use of the term "convex" to describe the rounded out bevel. Again, "out". "Out" from what? "Convex" is defined as out from the correlated flat. "Out" cannot be "in" at the same time in the same context. If your convex is thinner than your flat grind, then they were produced at different angles of incidence and
do not correlate. You might as well correlate a thinner flat grind with a thicker one and then state: "Look, this one is thinner!" Of course it is thinner, you sharpened it at a lower angle.
However, you can alter the shape of the bevel
without changing the angle of incidence, shoulder width, or bevel height by using a curved or flexible hone instead of a solid hone. How much the shape is altered is controlled by the amount of deformation and curvature (again, away from flat) of the hone. The result is a
thicker bevel, one with more metal that it would have if ground flat
at the same angle of incidence. Returning to the first diagram (C), the convex apex (violet) will always be more obtuse than the correlated flat apex (gray), which is the entire point (pun intended) - a more robust edge for a given bevel height & thickness. One CANNOT thin from a flat bevel to a convex bevel without widening the bevel, i.e. establishing an entirely different bevel. Conversely, one CAN thin from a convex edge to a flat-edge while maintaining the same bevel dimensions, reducing the apex angle.
In practice, if you want a thinner edge, widen the bevel - lowering the spine-to-hone distance accomplishes this (creating a lower apex-angle). If you want a more robust edge, EITHER reduce the bevel height (raising the spine-to-hone distance to a more obtuse sharpening angle)
OR use a flexible hone to sharpen convex and
maintain the same bevel height (same spine-to-hone distance).
