Linear isoquant [perfect substitutability of factors of production], Input-output isoquant or Leontif isoquant [no substitution or strict complementarity; only one efficient method of production] are exceptions to isoquant convexity to the origin. Kinked isoquant is of limited substitutability at kinks. But if kinks come closer and closer, it will become a smooth curve, convex to the origin.

negative slope, convexity to its origin

show how the price elasticity of demand is graphically measured along a liner demand curve?

indifference curve is a combination of two commodities. where as, isoquant curve shows a relationship between of variable factor i.e. labour and fixed factor i.e. capital.

1. Linear Isoquant: This type assumes perfect substitutability of factors of production: a given commodity may be produced by using only capital, or only labour, or by an infinite combination of K and L.
2. Input-Output Isoquant: This assumes strict complementarity[that is, zero substitutability] of the factors of production. The isoquant take the shape of a right angle. This type of isoquant is also called 'Leontief isoquant' after Leontief, who invented the input-output ananlysis.
3. Kinked Isoquant: This assmes limited substitutability of K and L. There are only a few processes for producing any one commodity. Substitutability of factors is possibleonly at the kinks. This form is also called 'activity analysis-isoquant' or 'linear-programming isoquant', because it is basically used in linear programming.
4. Smooth , Convex Isoquant: This form assumes continuous substitutability of K and L only over a certain range, beyond which factors cannot substitute each other. The isoquant appears as a smooth curve convex to the origin.