Higher order asymptotic powers of one-sided and two-sided tests, both conditional and unconditional, are evaluated for a one-parameter curved exponential family of distributions by using differential-geometrical notions. Higher order asymptotic powers and the expected size of one-sided and two-sided interval estimators, both conditional and unconditional, are also obtained. The tests and interval estimators, which are third-order most powerful at any arbitrary specified one point are explicitly designed with the help of the approximate ancillary statistic. The characteristics of the conditional inference are elucidated, by proving the equivalence up to the third order of the conditional inference and likelihood ratio inference. Geometrical notions such as curvatures and angles play a fundamental role in the present theory.