Although coarticulation frequently causes insertion and deletion errors in speech recognition systems, human beings can recognize it correctly, presumably because of correct estimation of glide between phonemes. The present study is concerned with the glide characteristics of vowels, and it is shown that the trajectory of the glide appears along geodesics calculated by the metric on a phonetical vowel space. First the spectrum envelope of the stable part of a vowel is digitized by representing it by a 40-dimensional vector. Then, this vector is projected to a three-dimensional parameter space by discriminant analysis. The trajectory of glide generally produced a curve, which suggests that the space is non-Euclidean, as the trajectory should be the shortest path in the space. Assuming that the distribution of each vowel is a normal distribution, it is shown that it is an ellipsoid with a covariance matrix. Second, by a transformation using metrics, based on the variance of vowels, the trajectory is transformed into Euclidean space. It is shown that the transformed trajectory of the vowel glide is a similar straight line.