Cartesian space path planning involves generating the position and orientation trajectories for a manipulator end-effector. Currently, much of the literature in motion planning for robotics concentrates on topics such as obstacle avoidance, dynamic optimizations, or high-level task planning. The focus of this research is on operator-generated motions. This will involve analytically studying the effects of higher-order properties (such as curvature and torsion) on the shape of spatial Cartesian curves. A particular emphasis will be placed on developing physical meanings and graphical visualization for these properties to aid the operator in generating geometrically complex motions. This research begins with a brief introduction to the domain of robotics and manipulator motion planning. An overview of work in the area of manipulator motion planning will demonstrate a lack of research on generating geometrically complex spatial paths. To pursue this goal, this report will then provide a review of the theory of algrebraic curves and their higher-order properties. This involves an evaluation of several different representations for both planar and spatial curves. Then, a survey of interactive curve generation techniques will be performed, which will draw from fields outside of robotics such as Computer Graphics and Computer-Aided Design (CAD). In addition to the reviewed methods, a new method for describing and generating spatial curves is proposed and demonstrated. This method begins with the study of a finite set of local geometric motion shapes (circular arcs, cusps, helices, etc). The local geometric shapes are studied in terms of their geometric parameters (curvature and torsion), analyzed to give physical meaning to these parameters, and displayed graphically as a family of curves based on these controlling parameters. This leads to the development of path constraints with well-defined physical meaning. Then, a curve generation method is developed that can convert these geometric constraints into parametric constraints and blend between them to form a complete motion program (cycle) of smooth paths connecting several carefully developed local curve properties. Up to ten distinct local curve shapes were developed in detail and one curve cycle demonstrated how all this could be combined into a full path planning scenario. Finally, the developed methods are packaged together into existing software and applied to an example demonstration.