In this paper the following two theorems are proved; 1) All intermediate joint axes of a robot arm with arbitrary number of joints intersect an extreme distance line between an arbitrary base point and the center point of the hand (extreme reach), and 2) all intermediate joint axes intersect an extreme perpendicular distance line from the center point of the hand to any arbitrary line in space. (The theorem for extreme mutual perpendicular distance lines has already been proved by Shimano and Roth [1, 2]). Using the first theorem an algorithm is developed for searching for extreme reaches. This algorithm can be applied with minor modifications to determine both extreme perpendicular lines and extreme mutual perpendicular lines. The algorithm can fail when applied to robot arms with special dimensions (for instance, robot arms with intersecting or parallel axes). Such cases are studied in detail in a second paper [3].

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