Intravascular brachytherapy (IVB) can significantly reduce the risk of restenosis after interventional treatment of stenotic arteries, if planned and applied correctly. To facilitate computer-based IVB planning, a three-dimensional vessel model has been derived from information on coronary artery segments acquired by intravascular ultrasound (IVUS) and biplane angiography. Part I describes the approach of model construction and presents possibilities of visualization. The vessel model is represented by a voxel volume. Polygonal information about the vessel wall structure is derived by segmentation from a sequence of IVUS images automatically acquired ECG gated during pull back of the IVUS transducer. To detect horizontal, vertical, and radial contours, modified Canny-Edge and Shen-Castan filters are applied on Cartesian and polar coordinate representations of the IVUS tomograms as edge detectors. The spatial course of the vessel wall layers is traced in reconstructed longitudinal IVUS scans. By resampling the sequence of IVUS frames the voxel volume is obtained. For this purpose the frames are properly located in space and augmented with additional intermediate frames generated by interpolation. Their spatial location and orientation is derived from biplane X-ray angiography which is performed simultaneously. For resampling, two approaches are proposed: insertion of the vertices of the rectangular goal grid into the cells of a deformed hexahedral mesh derived from the IVUS sequence, and insertion of the vertices of the hexahedral mesh into the cells of the rectangular grid. Finally, the vessel model is visualized by methods of combined volume and polygon rendering. The segmentation process is verified as being in good agreement with results obtained by manual contour tracing with a commercial system. Our approach of construction of the vessel model has been implemented into an interactive software system, 3D IVUS-View, serving as the basis of a future system for intracoronary brachytherapy treatment planning being currently under development (Part II).