spur gear vs rack and pinion design

Yes.

In fact, standard gear hobbing uses a rack-shaped cutter to machine spur gears. It's the magic of the involute form that any diameter gear (including the infinite-diameter rack) will work.

Hth, Fred Klingener

then how can we calculate number of teeth for rack

as for spur gear number of teeth = diameter/ module

but for rack we have no diameter

A rack is a gear with infinite (arbitrarily large) diameter.

Ned Simmons

Talk about the tooth pitch instead.

Tooth pitch is the developed length between the teeth along the pitch circumference. On a spur (pinion) gear, it's a circular arc. On a rack, it's a straight line.

Spur gear pairs roll on tangent pitch circles. Pitch circle of the pinion rolls on the rack pitck line.

Hth, Fred Klingener

if number of teeth on a spur is pitch diameter/module Then tooth spacing on the pitch circle is pitch circle circumference / number of teeth or pi X D / (D/module) = pi X module. This is evidently the tooth pitch you want on the rack.

Brian Whatcott Altus OK