A vibration cutting insert includes a top surface having a rake face and a breaker face, a first side surface, a second side surface, and a bottom surface. In a cross section which includes a bisector of an angle formed between a first ridgeline and a second ridgeline when viewed in a direction from the top surface toward the bottom surface, and which is parallel to the direction from the top surface toward the bottom surface, a rake angle formed between the rake face and a plane parallel to the bottom surface has a positive angle. The angle formed between the first ridgeline and the second ridgeline is an acute angle. The rake face constitutes a cylindrical surface. The breaker face constitutes a flat surface.

A vibration cutting insert includes a top surface having a rake face and a breaker face, a first side surface, a second side surface, and a bottom surface. In a cross section which includes a bisector of an angle formed between a first ridgeline and a second ridgeline when viewed in a direction from the top surface toward the bottom surface, and which is parallel to the direction from the top surface toward the bottom surface, a rake angle formed between the rake face and a plane parallel to the bottom surface has a positive angle. The angle formed between the first ridgeline and the second ridgeline is an acute angle. The rake face constitutes a cylindrical surface. The breaker face constitutes a flat surface.

A controller controls a machine tool, which performs thread cutting for a workpiece, according to a machining program. The controller analyzes operating conditions of a cutting-up/cutting-in motion commanded in the machining program and inserts a cutting-up/cutting-in motion created based on the analyzed operating conditions into the thread cutting. In the insertion of the cutting-up/cutting-in motion into the thread cutting, a cycle including a cutting-in operation, a cutting-up operation for separating chips by raising the cutting tool for cutting in the radial direction of the workpiece, and an operation to cause the cutting tool to approach a start position of an immediately preceding cutting-up operation so as not to interfere with the workpiece is repeatedly performed.

A controller controls a machine tool, which performs thread cutting for a workpiece, according to a machining program. The controller analyzes operating conditions of a cutting-up/cutting-in motion commanded in the machining program and inserts a cutting-up/cutting-in motion created based on the analyzed operating conditions into the thread cutting. In the insertion of the cutting-up/cutting-in motion into the thread cutting, a cycle including a cutting-in operation, a cutting-up operation for separating chips by raising the cutting tool for cutting in the radial direction of the workpiece, and an operation to cause the cutting tool to approach a start position of an immediately preceding cutting-up operation so as not to interfere with the workpiece is repeatedly performed.

A face grooving tool body includes a blade portion having a top surface and an opposite bottom surface, a first side surface and an opposite second side surface, and a front end and an opposite rear end. A longitudinal axis of the blade portion coincides with a primary cutting feed direction of the blade portion. An upper blade portion is associated with the top surface and a lower blade portion is associated with the bottom surface, the upper and lower blade portions being separated by a slit. The second side surface of the lower blade portion is curved around a second side surface axis of curvature parallel to the longitudinal axis. A key hole is formed in the blade portion. The slit includes a slit portion that has the shape of a concave function in a side view.

A tool having a cutting edge that includes a sintered body containing cubic boron nitride. The sintered body integrally and inseparably includes an inner region and a binder phase enriched layer formed on at least part of a surface of the inner region. The inner region includes: 15-90 volume % of cubic boron nitride; and 10-85 volume % of a mixture of a binder phase and impurities. The binder phase enriched layer includes: 90-100 volume % of the binder phase and impurities mixture; and 0-10 volume % of cubic boron nitride; and the binder phase contains at least one kind selected from the group consisting of: at least one element selected from the group consisting of Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W, Al, Co, Ni and Si; and a compound of the element and at least one element selected from the group consisting of C, N, O and B.

A grooving insert includes a front surface and an opposite rear surface, a first side surface and an opposite second side surface, and a top surface and an opposite bottom surface. The top surface includes a top front portion and a top rear portion, the top front portion having a cutting edge arranged symmetrically relative to a first plane. The bottom surface is symmetrically arranged in relation to a second plane, the first plane and the second plane forming an angle () of 0.5-5.0 relative to each other.

The edge portion includes an upper surface, a side surface, and a land surface. The side surface has a front side surface and a pair of lateral side surfaces. An intersection between the land surface and the front side surface forms a front cutting edge. An intersection between the land surface and each of the pair of lateral side surfaces form a corresponding one of a pair of lateral cutting edges. The edge portion contains 80 vol % or more of diamond. A chip breaker recess is provided between the upper surface and the land surface. The surfaces forming the chip breaker recess include a rake face and a breaker wall surface. The upper surface has a front edge portion opposite to the front cutting edge from the chip breaker recess. A pair of protruding portions are provided to extend from the front edge portion toward the front cutting edge.

The edge portion includes an upper surface, a side surface, and a land surface. The side surface has a front side surface and a pair of lateral side surfaces. An intersection between the land surface and the front side surface forms a front cutting edge. An intersection between the land surface and each of the pair of lateral side surfaces form a corresponding one of a pair of lateral cutting edges. The edge portion contains 80 vol % or more of diamond. A chip breaker recess is provided between the upper surface and the land surface. The surfaces forming the chip breaker recess include a rake face and a breaker wall surface. The upper surface has a front edge portion opposite to the front cutting edge from the chip breaker recess. A pair of protruding portions are provided to extend from the front edge portion toward the front cutting edge.

A cutting insert includes an upper surface and upper edge. The upper surface includes a second side part, a first corner part, and a second corner part. The upper surface further includes a first surface and a second surface. The upper edge includes a first edge located at the first corner part, a second edge located at the second side part, and a third edge located at the second corner part. The second surface includes a first region extending toward the first corner part, and a second region extending toward the second corner part. A distance between the first region and the upper edge is smaller at a side of the second edge than at a side of the first edge. A distance between the second region and the upper edge is smaller at a side of the second edge than at a side of the third edge.